Question 1 1 of 269 selected Lines, Angles, & Triangles
In the figure, . The measure of angle is , and the measure of angle is . What is the value of ?
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Correct Answer: 83The correct answer is . It's given that in the figure, . Thus, triangle is an isosceles triangle and the measure of angle is equal to the measure of angle . The sum of the measures of the interior angles of a triangle is . Thus, the sum of the measures of the interior angles of triangle is . It's given that the measure of angle is . It follows that the sum of the measures of angles and is , or . Since the measure of angle is equal to the measure of angle , the measure of angle is half of , or . The sum of the measures of the interior angles of triangle is . It's given that the measure of angle is . Since the measure of angle , which is the same angle as angle , is , it follows that the measure of angle is , or . Since angle and angle form a straight line, the sum of the measures of these angles is . It's given in the figure that the measure of angle is . It follows that . Subtracting from both sides of this equation yields .
Question 2 2 of 269 selected Lines, Angles, & Triangles
Triangles and are congruent, where , , and correspond to , , and , respectively. The measure of angle is and the measure of angle is . What is the measure of angle ?
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Correct Answer: BChoice B is correct. It's given that triangles and are congruent such that angle corresponds to angle . Corresponding angles of congruent triangles are congruent, so angle and angle must be congruent. Therefore, if the measure of angle is , then the measure of angle is also .
Choice A is incorrect. This is the measure of angle , not angle .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 3 3 of 269 selected Area & Volume
The length of each edge of a box is inches. Each side of the box is in the shape of a square. The box does not have a lid. What is the exterior surface area, in square inches, of this box without a lid?
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Correct Answer: 4205The correct answer is . The exterior surface area of a figure is the sum of the areas of all its faces. It's given that the box does not have a lid and that each side of the box is in the shape of a square. Therefore, the box consists of congruent square faces. It's also given that the length of each edge is inches. Let represent the length of an edge of a square. It follows that the area of a square is equal to . Therefore, the area of each of the square faces is equal to , or , square inches. Since the box consists of congruent square faces, it follows that the sum of the areas of all its faces, or the exterior surface area of this box without a lid, is , or , square inches.
Question 4 4 of 269 selected Lines, Angles, & Triangles
In the figure shown, points , , , and lie on line segment , and line segment intersects line segment at point . The measure of is , the measure of is , the measure of is , and the measure of is . What is the measure, in degrees, of ?
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Correct Answer: 123The correct answer is . The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is degrees. It's given that the measure of is and the measure of is . Since points , , and form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of is , or . It's also given that the measure of is . Since and are supplementary angles, the sum of their measures is degrees. It follows that the measure, in degrees, of is , or . Since points , , and form a triangle, and is the same angle as , it follows from the triangle angle sum theorem that the measure, in degrees, of is , or . It's given that the measure of is . Since and are supplementary angles, the sum of their measures is degrees. It follows that the measure, in degrees, of is , or . Since points , , and form a triangle, and is the same angle as , it follows from the triangle angle sum theorem that the measure, in degrees, of is , or .
Question 5 5 of 269 selected Right Triangles & Trigonometry
Triangle is similar to triangle , where angle corresponds to angle and angles and are right angles. If , what is the value of ?
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Correct Answer: BChoice B is correct. If two triangles are similar, then their corresponding angles are congruent. It's given that right triangle is similar to right triangle and angle corresponds to angle . It follows that angle is congruent to angle and, therefore, the measure of angle is equal to the measure of angle . The sine ratios of angles of equal measure are equal. Since the measure of angle is equal to the measure of angle , . It's given that . Therefore, is .
Choice A is incorrect. This is the value of , not the value of .
Choice C is incorrect. This is the reciprocal of the value of , not the value of .
Choice D is incorrect. This is the reciprocal of the value of , not the value of .
Question 6 6 of 269 selected Lines, Angles, & Triangles
Intersecting lines r, s, and t are shown below.
What is the value of x ?
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The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of
and the other, which is supplementary to the angle with measure
, has measure of
. Therefore, the value of x is
.
Question 7 7 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line . What is the value of ?
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Correct Answer: DChoice D is correct. It's given that lines and are parallel. Since line intersects both lines and , it's a transversal. The angles in the figure marked as and are on the same side of the transversal, where one is an interior angle with line as a side, and the other is an exterior angle with line as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 8 8 of 269 selected Area & Volume
A right circular cylinder has a volume of . If the height of the cylinder is 5, what is the radius of the cylinder?
3
4.5
9
40
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Correct Answer: AChoice A is correct. The volume of a right circular cylinder with a radius of r is the product of the area of the base, , and the height, h. The volume of the right circular cylinder described is
and its height is 5. If the radius is r, it follows that
. Dividing both sides of this equation by
yields
. Taking the square root of both sides yields
or
. Since r represents the radius, the value must be positive. Therefore, the radius is 3.
Choice B is incorrect and may result from finding that the square of the radius is 9, but then from dividing 9 by 2, rather than taking the square root of 9. Choice C is incorrect. This represents the square of the radius. Choice D is incorrect and may result from solving the equation for
, not r, by dividing by
on both sides and then by subtracting, not dividing, 5 from both sides.
Question 9 9 of 269 selected Area & Volume
Square A has side lengths that are times the side lengths of square B. The area of square A is times the area of square B. What is the value of ?
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Correct Answer: 27556The correct answer is . The area of a square is , where is the side length of the square. Let represent the length of each side of square B. Substituting for in yields . It follows that the area of square B is . It’s given that square A has side lengths that are times the side lengths of square B. Since represents the length of each side of square B, the length of each side of square A can be represented by the expression . It follows that the area of square A is , or . It’s given that the area of square A is times the area of square B. Since the area of square A is equal to , and the area of square B is equal to , an equation representing the given statement is . Since represents the length of each side of square B, the value of must be positive. Therefore, the value of is also positive, so it does not equal . Dividing by on both sides of the equation yields . Therefore, the value of is .
Question 10 10 of 269 selected Right Triangles & Trigonometry
In right triangle , the sum of the measures of angle and angle is degrees. The value of is . What is the value of ?
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Correct Answer: BChoice B is correct. The sine of any acute angle is equal to the cosine of its complement. It’s given that in right triangle , the sum of the measures of angle and angle is degrees. Therefore, angle and angle are complementary, and the value of is equal to the value of . It's given that the value of is , so the value of is also .
Choice A is incorrect. This is the value of .
Choice C is incorrect. This is the value of .
Choice D is incorrect. This is the value of .
Question 11 11 of 269 selected Area & Volume
What is the area, in square inches, of a rectangle with a length of inches and a width of inches?
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Correct Answer: CChoice C is correct. The area, , of a rectangle is given by the formula , where represents the length of the rectangle and represents its width. It’s given that the rectangle has a length of inches and a width of inches. Substituting for and for in the formula yields , or . Thus, the area, in square inches, of the rectangle is .
Choice A is incorrect. This is the sum, not the product, of the length and width of the rectangle.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is twice the area, in square inches, of the rectangle.
Question 12 12 of 269 selected Area & Volume
The side length of a square is . What is the area, , of the square?
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Correct Answer: CChoice C is correct. The area , , of a square with side length , , is given by the formula . It’s given that the square has a side length of . Substituting for in the formula yields , or . Therefore, the area, , of the square is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the perimeter, , of the square, not its area, .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 13 13 of 269 selected Area & Volume
Triangle R has an area of . Square S has side lengths of . What is the total area of triangle R and square S, ?
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Correct Answer: DChoice D is correct. It’s given that triangle R has an area of . The area of a square is , where is the side length of the square. It's given that square S has side lengths of . It follows that the area, in , of square S is , or . Therefore, the total area, in , of triangle R and square S is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 14 14 of 269 selected Area & Volume
What is the length of one side of a square that has the same area as a circle with radius 2 ?
2
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Correct Answer: CChoice C is correct. The area A of a circle with radius r is given by the formula . Thus, a circle with radius 2 has area
, which can be rewritten as
. The area of a square with side length s is given by the formula
. Thus, if a square has the same area as a circle with radius 2, then
. Since the side length of a square must be a positive number, taking the square root of both sides of
gives
. Using the properties of square roots,
can be rewritten as
, which is equivalent to
. Therefore,
.
Choice A is incorrect. The side length of the square isn’t equal to the radius of the circle. Choices B and D are incorrect and may result from incorrectly simplifying the expression .
Question 15 15 of 269 selected Area & Volume
A right square prism has a height of units. The volume of the prism is cubic units. What is the length, in units, of an edge of the base?
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Correct Answer: 12The correct answer is . The volume, , of a right square prism can be calculated using the formula , where represents the length of an edge of the base and represents the height of the prism. It’s given that the volume of the prism is cubic units and the height is units. Substituting for and for in the formula yields . Dividing both sides of this equation by yields . Taking the square root of both sides of this equation yields two solutions: and . The length can't be negative, so . Therefore, the length, in units, of an edge of the base is .
Question 16 16 of 269 selected Right Triangles & Trigonometry
In triangle , angle is a right angle and the length of is units. If , what is the perimeter, in units, of triangle ?
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Correct Answer: BChoice B is correct. It's given that angle in triangle is a right angle. Thus, side is the leg opposite angle and side is the leg adjacent to angle . The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. It follows that . It's given that and the length of side is units. Substituting for and for in the equation yields . Multiplying both sides of this equation by yields , or . Dividing both sides of this equation by yields . The length can be calculated using the Pythagorean theorem, which states that if a right triangle has legs with lengths of and and a hypotenuse with length , then . Substituting for and for in this equation yields , or . Taking the square root of both sides of this equation yields . Since the length of the hypotenuse must be positive, . Therefore, the length of is units. The perimeter of a triangle is the sum of the lengths of all sides. Thus, units, or units, is the perimeter of triangle .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This would be the perimeter, in units, for a right triangle where the length of side is units, not units.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 17 17 of 269 selected Right Triangles & Trigonometry
Triangle shown is a right triangle. Which of the following has the same value as ?
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Correct Answer: DChoice D is correct. The sine of an angle is equal to the cosine of its complementary angle. In the triangle shown, angle is a right angle; thus, angles and are complementary angles. Therefore, has the same value as .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Question 18 18 of 269 selected Lines, Angles, & Triangles
In the given triangle, and
has a measure of
. What is the value of x ?
36
46
58
70
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Correct Answer: BChoice B is correct. Since , the measures of their corresponding angles,
and
, are equal. Since
has a measure of
, the measure of
is also
. Since the sum of the measures of the interior angles in a triangle is
, it follows that
, or
. Subtracting by 134 on both sides of this equation yields
.
Choices A, C, and D are incorrect and may result from calculation errors.
Question 19 19 of 269 selected Lines, Angles, & Triangles
In the figure above, and
intersect at point P,
, and
. What is the measure, in degrees, of
? (Disregard the degree symbol when gridding your answer.)
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The correct answer is 30. It is given that the measure of is
. Angle MPR and
are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is
, and so the measure of
is
. The sum of the angles in a triangle is
. Subtracting the measure of
from
yields the sum of the other angles in the triangle MPR. Since
, the sum of the measures of
and
is
. It is given that
, so it follows that triangle MPR is isosceles. Therefore
and
must be congruent. Since the sum of the measure of these two angles is
, it follows that the measure of each angle is
.
An alternate approach would be to use the exterior angle theorem, noting that the measure of is equal to the sum of the measures of
and
. Since both angles are equal, each of them has a measure of
.
Question 20 20 of 269 selected Area & Volume
Square X has a side length of centimeters. The perimeter of square Y is times the perimeter of square X. What is the length, in centimeters, of one side of square Y?
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Correct Answer: DChoice D is correct. The perimeter, , of a square can be found using the formula , where is the length of each side of the square. It's given that square X has a side length of centimeters. Substituting for in the formula for the perimeter of a square yields , or . Therefore, the perimeter of square X is centimeters. It’s also given that the perimeter of square Y is times the perimeter of square X. Therefore, the perimeter of square Y is , or , centimeters. Substituting for in the formula gives . Dividing both sides of this equation by gives . Therefore, the length of one side of square Y is centimeters.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 21 21 of 269 selected Area & Volume
The circumference of the base of a right circular cylinder is meters, and the height of the cylinder is meters. What is the volume, in cubic meters, of the cylinder?
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Correct Answer: CChoice C is correct. The volume, , of a right circular cylinder is given by the formula , where is the radius of the base of the cylinder and is the height of the cylinder. It’s given that a right circular cylinder has a height of meters. Therefore, . It's also given that the right circular cylinder has a base with a circumference of meters. The circumference, , of a circle is given by , where is the radius of the circle. Substituting for in the formula yields . Dividing each side of this equation by yields . Substituting for and for in the formula yields , or . Therefore, the volume, in cubic meters, of the cylinder is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the lateral surface area, not the volume, of the cylinder.
Choice D is incorrect. This is the result of using the diameter, not the radius, for the value of in the formula .
Question 22 22 of 269 selected Lines, Angles, & Triangles
Right triangles and are similar, where corresponds to . If the measure of angle is , what is the measure of angle ?
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Correct Answer: BChoice B is correct. In similar triangles, corresponding angles are congruent. It’s given that right triangles and are similar, where angle corresponds to angle . It follows that angle is congruent to angle . In the triangles shown, angle and angle are both marked as right angles, so angle and angle are corresponding angles. It follows that angle and angle are corresponding angles, and thus, angle is congruent to angle . It’s given that the measure of angle is , so the measure of angle is also . Angle is a right angle, so the measure of angle is . The sum of the measures of the interior angles of a triangle is . Thus, the sum of the measures of the interior angles of triangle is degrees. Let represent the measure, in degrees, of angle . It follows that , or . Subtracting from both sides of this equation yields . Therefore, if the measure of angle is degrees, then the measure of angle is degrees.
Choice A is incorrect. This is the measure of angle .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the sum of the measures of angle and angle .
Question 23 23 of 269 selected Area & Volume
In the figure shown, triangle is similar to triangle , where corresponds to and corresponds to . The length of is , and the perimeter of triangle is . The length of is . What is the perimeter of triangle ?
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Correct Answer: AChoice A is correct. It’s given that triangle is similar to triangle , where corresponds to and corresponds to . It follows that corresponds to . If two triangles are similar, then the scale factor between their perimeters is equal to the scale factor between the lengths of their corresponding sides. It's given that the length of is and the length of is . Therefore, the scale factor from the length of to the length of is , or . It’s given that the perimeter of triangle is . Let represent the perimeter of triangle . It follows that . Multiplying each side of this equation by yields . Therefore, the perimeter of triangle is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 24 24 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line , and both lines are intersected by line . If , what is the value of ?
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Correct Answer: 57The correct answer is . Based on the figure, the angle with measure and the angle vertical to the angle with measure are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure also has measure . It’s given that lines and are parallel. Therefore, same side interior angles between lines and are supplementary. It follows that . If , then the value of can be found by substituting for in the equation , which yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Thus, if , the value of is .
Question 25 25 of 269 selected Lines, Angles, & Triangles
In the figure above, lines and m are parallel,
, and
. What is the value of x ?
120
100
90
80
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Correct Answer: BChoice B is correct. Let the measure of the third angle in the smaller triangle be . Since lines
and m are parallel and cut by transversals, it follows that the corresponding angles formed are congruent. So
. The sum of the measures of the interior angles of a triangle is
, which for the interior angles in the smaller triangle yields
. Given that
and
, it follows that
. Solving for x gives
, or
.
Choice A is incorrect and may result from incorrectly assuming that angles . Choice C is incorrect and may result from incorrectly assuming that the smaller triangle is a right triangle, with x as the right angle. Choice D is incorrect and may result from a misunderstanding of the exterior angle theorem and incorrectly assuming that
.
Question 26 26 of 269 selected Lines, Angles, & Triangles
In the figure above, ,
, and
are parallel. Points B and E lie on
and
, respectively. If
, and
, what is the length of
, to the nearest tenth?
16.8
17.5
18.4
19.6
Show Answer
Correct Answer: BChoice B is correct. Since ,
, and
are parallel, quadrilaterals
and
are similar. Let x represent the length of
. With similar figures, the ratios of the lengths of corresponding sides are equal. It follows that
. Multiplying both sides of this equation by 18.5 and by x yields
, or
. Dividing both sides of this equation by 9 yields
, which to the nearest tenth is 17.5.
Choices A, C, and D are incorrect and may result from errors made when setting up the proportion.
Question 27 27 of 269 selected Area & Volume
The area of a square is square inches. What is the side length, in inches, of this square?
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Correct Answer: AChoice A is correct. It's given that the area of a square is square inches. The area , in square inches, of a square is given by the formula , where is the side length, in inches, of the square. Substituting for in this formula yields . Taking the positive square root of both sides of this equation yields . Thus, the side length, in inches, of this square is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the area, in square inches, of the square, not the side length, in inches, of the square.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 28 28 of 269 selected Right Triangles & Trigonometry
In the figure above, what is the value of ?
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Correct Answer: CChoice C is correct. Angle A is an acute angle in a right triangle, so the value of tan(A) is equivalent to the ratio of the length of the side opposite angle A, 20, to the length of the nonhypotenuse side adjacent to angle A, 21. Therefore, .
Choice A is incorrect. This is the value of sin(A). Choice B is incorrect. This is the value of cos(A). Choice D is incorrect. This is the value of tan(B).
Question 29 29 of 269 selected Area & Volume
A circle has a radius of meters. What is the area, in square meters, of the circle?
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Correct Answer: DChoice D is correct. The area, , of a circle is given by the formula , where is the radius of the circle. It’s given that the circle has a radius of meters. Substituting for in the formula yields , or . Therefore, the area, in square meters, of the circle is .
Choice A is incorrect. This is the area, in square meters, of a circle with a radius of meters.
Choice B is incorrect. This is the area, in square meters, of a circle with a radius of meters.
Choice C is incorrect. This is the circumference, in meters, of the circle.
Question 30 30 of 269 selected Circles
A circle in the xy-plane has its center at . Line is tangent to this circle at the point . Which of the following points also lies on line ?
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Correct Answer: CChoice C is correct. It’s given that the circle has its center at and that line is tangent to this circle at the point . Therefore, the points and are the endpoints of the radius of the circle at the point of tangency. The slope of a line or line segment that contains the points and can be calculated as . Substituting for and for in the expression yields , or . Thus, the slope of this radius is . A line that’s tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It follows that line is perpendicular to the radius at the point , so the slope of line is the negative reciprocal of the slope of this radius. The negative reciprocal of is . Therefore, the slope of line is . Since the slope of line is the same between any two points on line , a point lies on line if the slope of the line segment connecting the point and is . Substituting choice C, , for and for in the expression yields , or . Therefore, the point lies on line .
Choice A is incorrect. The slope of the line segment connecting and is , or , not .
Choice B is incorrect. The slope of the line segment connecting and is , or , not .
Choice D is incorrect. The slope of the line segment connecting and is , or , not .
Question 31 31 of 269 selected Circles
Points and lie on a circle with center . The radius of this circle is inches. Triangle has a perimeter of inches. What is the length, in inches, of ?
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Correct Answer: BChoice B is correct. Since it's given that is the center of a circle with a radius of inches, and that points and lie on that circle, it follows that and of triangle each have a length of inches. Let the length of be inches. It follows that the perimeter of triangle is inches. Since it's given that the perimeter of triangle is inches, it follows that , or . Subtracting from both sides of this equation gives . Therefore, the length, in inches, of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 32 32 of 269 selected Area & Volume
The three points shown define a circle. The circumference of this circle is , where is a constant. What is the value of ?
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Correct Answer: BChoice B is correct. It’s given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point is halfway between the points and and is a distance of units from each of those two points. The point is also a distance of units from . Because the point is the same distance from all three given points, it must be the center of the circle. The radius of a circle is the distance from the center to any point on the circle. Since that distance is , it follows that the radius of the circle is . The circumference of a circle with radius is equal to . It follows that the circumference of the circle is , or . It's given that the circumference of the circle is . Therefore, the value of is .
Choice A is incorrect. This is the radius of the circle, not the value of in the expression .
Choice C is incorrect. This is the x-coordinate of the center of the circle, not the value of in the expression .
Choice D is incorrect. This is the value of for which represents the area of the circle, in square units, not the circumference of the circle, in units.
Question 33 33 of 269 selected Area & Volume
What is the volume, in cubic centimeters, of a right rectangular prism that has a length of 4 centimeters, a width of 9 centimeters, and a height of 10 centimeters?
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The correct answer is 360. The volume of a right rectangular prism is calculated by multiplying its dimensions: length, width, and height. Multiplying the values given for these dimensions yields a volume of cubic centimeters.
Question 34 34 of 269 selected Area & Volume
A cube has an edge length of inches. A solid sphere with a radius of inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?
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Correct Answer: AChoice A is correct. The volume of a cube can be found by using the formula , where is the volume and is the edge length of the cube. Therefore, the volume of the given cube is , or cubic inches. The volume of a sphere can be found by using the formula , where is the volume and is the radius of the sphere. Therefore, the volume of the given sphere is , or approximately cubic inches. The volume of the space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere. Subtracting the approximate volume of the sphere from the volume of the cube gives cubic inches.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 35 35 of 269 selected Area & Volume
A right rectangular prism has a length of meters, a width of meters, and a height of meters. What is the volume, in cubic meters, of the prism?
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Correct Answer: 880The correct answer is . The volume, , of a right rectangular prism is given by the formula , where is the length, is the width, and is the height of the prism. It’s given that a right rectangular prism has a length of meters, a width of meters, and a height of meters. Substituting for , for , and for in the formula yields , or . Therefore, the volume, in cubic meters, of the prism is .
Question 36 36 of 269 selected Lines, Angles, & Triangles
In triangle , the measure of angle is and is an altitude of the triangle. The length of is and the length of is greater than the length of . What is the value of ?
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Correct Answer: DChoice D is correct. It's given that in triangle , the measure of angle is and is an altitude of the triangle. Therefore, the measure of angle is . It follows that angle is congruent to angle and angle is congruent to angle . By the angle-angle similarity postulate, triangle is similar to triangle . Since triangles and are similar, it follows that . It's also given that the length of is and the length of is greater than the length of . Therefore, the length of is , or . Substituting for and for in the equation yields . Therefore, the value of is .
Choice A is incorrect. This is the value of .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 37 37 of 269 selected Area & Volume
The figure shown is a right circular cylinder with a radius of and height of . A second right circular cylinder (not shown) has a volume that is times as large as the volume of the cylinder shown. Which of the following could represent the radius , in terms of , and the height , in terms of , of the second cylinder?
and
and
and
and
Show Answer
Correct Answer: CChoice C is correct. The volume of a right circular cylinder is equal to , where is the radius of a base of the cylinder and is the height of the cylinder. It’s given that the cylinder shown has a radius of and a height of . It follows that the volume of the cylinder shown is equal to . It’s given that the second right circular cylinder has a radius of and a height of . It follows that the volume of the second cylinder is equal to . Choice C gives and . Substituting for and for in the expression that represents the volume of the second cylinder yields , or , which is equivalent to , or . This expression is equal to times the volume of the cylinder shown, . Therefore, and could represent the radius , in terms of , and the height , in terms of , of the second cylinder.
Choice A is incorrect. Substituting for and for in the expression that represents the volume of the second cylinder yields , or , which is equivalent to , or . This expression is equal to , not , times the volume of the cylinder shown.
Choice B is incorrect. Substituting for and for in the expression that represents the volume of the second cylinder yields , or , which is equivalent to , or . This expression is equal to , not , times the volume of the cylinder shown.
Choice D is incorrect. Substituting for and for in the expression that represents the volume of the second cylinder yields , or , which is equivalent to , or . This expression is equal to , not , times the volume of the cylinder shown.
Question 38 38 of 269 selected Lines, Angles, & Triangles
In , the measure of is and the measure of is . What is the measure of ?
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Correct Answer: AChoice A is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is . It's given that in , the measure of is and the measure of is . It follows that the measure of is , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the sum of the measures of and , not the measure of .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 39 39 of 269 selected Area & Volume
A right circular cylinder has a volume of cubic centimeters. The area of the base of the cylinder is square centimeters. What is the height, in centimeters, of the cylinder?
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Correct Answer: 29The correct answer is . The volume, , of a right circular cylinder is given by the formula , where is the radius of the base of the cylinder and is the height of the cylinder. Since the base of the cylinder is a circle with radius , the area of the base of the cylinder is . It's given that a right circular cylinder has a volume of cubic centimeters; therefore, . It's also given that the area of the base of the cylinder is square centimeters; therefore, . Substituting for and for in the formula yields . Dividing both sides of this equation by yields . Therefore, the height of the cylinder, in centimeters, is .
Question 40 40 of 269 selected Lines, Angles, & Triangles
Two nearby trees are perpendicular to the ground, which is flat. One of these trees is feet tall and has a shadow that is feet long. At the same time, the shadow of the other tree is feet long. How tall, in feet, is the other tree?
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Correct Answer: BChoice B is correct. Each tree and its shadow can be modeled using a right triangle, where the height of the tree and the length of its shadow are the legs of the triangle. At a given point in time, the right triangles formed by two nearby trees and their respective shadows will be similar. Therefore, if the height of the other tree is , in feet, the value of can be calculated by solving the proportional relationship . This equation is equivalent to, or . Multiplying each side of the equation by yields . Therefore, the other tree is tall.
Choice A is incorrect and may result from calculating the difference between the lengths of the shadows, rather than the height of the other tree.
Choice C is incorrect and may result from calculating the difference between the height of the -foot-tall tree and the length of the shadow of the other tree, rather than calculating the height of the other tree.
Choice D is incorrect and may result from a conceptual or calculation error.
Question 41 41 of 269 selected Lines, Angles, & Triangles
is similar to . The lengths represented by , , , and in the figure are , , , and , respectively. What is the length of ?
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Correct Answer: AChoice A is correct. The figure shows that angle in and angle in are right angles. It follows that angle is congruent to angle . The figure also shows that the measures of angle and angle are both . Therefore, angle is congruent to angle . It’s given that is similar to . Since angle is congruent to angle , and angle is congruent to angle , it follows that corresponds to , and corresponds to . Since corresponding sides of similar triangles are proportional, it follows that . It’s also given that the lengths of , , and are , , and , respectively. Substituting for , for , and for in the equation yields . Multiplying each side of this equation by yields , or . Thus, the length of is .
Choice B is incorrect. This is the result of solving the equation , not .
Choice C is incorrect. This is the result of solving the equation , not .
Choice D is incorrect. This is the result of solving the equation , not .
Question 42 42 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line . What is the value of ?
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Correct Answer: DChoice D is correct. It's given that lines and are parallel. Since line intersects both lines and , it's a transversal. The angles in the figure marked as and are on the same side of the transversal, where one is an interior angle with line as a side, and the other is an exterior angle with line as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 43 43 of 269 selected Lines, Angles, & Triangles
Triangle is similar to triangle , where , , and correspond to , , and , respectively. In triangle , the length of is and the length of is . In triangle , the length of is . What is the length of ?
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Correct Answer: DChoice D is correct. It's given that triangle is similar to triangle , where , , and correspond to , , and , respectively. It follows that side corresponds to side and side corresponds to side . Since the lengths of corresponding sides in similar triangles are proportional, it follows that . Substituting for , for , and for in this equation yields . Multiplying each side of this equation by yields . Therefore, the length of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the length of , not .
Question 44 44 of 269 selected Lines, Angles, & Triangles
In the figure, , the measure of angle is , and the measure of angle is . What is the value of ?
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Correct Answer: 156The correct answer is . In the figure shown, the sum of the measures of angle and angle is . It’s given that the measure of angle is . Therefore, the measure of angle is , or . The sum of the measures of the interior angles of a triangle is . In triangle , the measure of angle is and it's given that the measure of angle is . Thus, the measure of angle is , or . It’s given that . Therefore, triangle is an isosceles triangle and the measure of is equal to the measure of angle . In triangle , the measure of angle is and the measure of angle is . Thus, the measure of angle is , or . The figure shows that the measure of angle is , so the value of is .
Question 45 45 of 269 selected Lines, Angles, & Triangles
In the figure shown, and intersect at point . , , , and . What is the length of ?
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Correct Answer: 54The correct answer is . The figure shown includes two triangles, triangle and triangle , such that angle and angle are vertical angles. It follows that angle is congruent to angle . It’s also given in the figure that the measures of angle and angle are . Therefore angle is congruent to angle . Since triangle and triangle have two pairs of congruent angles, triangle is similar to triangle by the angle-angle similarity postulate, where corresponds to , and corresponds to . Since the lengths of corresponding sides in similar triangles are proportional, it follows that . It’s given that , , and . Substituting for , for , and for in the equation yields . Multiplying each side of this equation by yields , or . Therefore, the length of is .
Question 46 46 of 269 selected Circles
The circle above has center O, the length of arc is
, and
. What is the length of arc
?
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Correct Answer: BChoice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is 360°, it follows that arc
is subtended by a central angle with measure
. If s is the length of arc
, then s must satisfy the ratio
. Reducing the fraction
to its simplest form gives
. Therefore,
. Multiplying both sides of
by
yields
.
Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc
, not its full length.
Question 47 47 of 269 selected Lines, Angles, & Triangles
In triangles and , angles and each have measure and angles and each have measure . Which additional piece of information is sufficient to determine whether triangle is congruent to triangle ?
The measure of angle
The length of side
The lengths of sides and
No additional information is necessary.
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Correct Answer: CChoice C is correct. Since angles and each have the same measure and angles and each have the same measure, triangles and are similar, where side corresponds to side . To determine whether two similar triangles are congruent, it is sufficient to determine whether one pair of corresponding sides are congruent. Therefore, to determine whether triangles and are congruent, it is sufficient to determine whether sides and have equal length. Thus, the lengths of and are sufficient to determine whether triangle is congruent to triangle .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect. The given information is sufficient to determine that triangles and are similar, but not whether they are congruent.
Question 48 48 of 269 selected Right Triangles & Trigonometry
What is the value of in the triangle shown?
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Correct Answer: BChoice B is correct. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to angle is , and the length of the hypotenuse is . Therefore, .
Choice A is incorrect. This is the value of .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 49 49 of 269 selected Right Triangles & Trigonometry
In triangle , angle is a right angle, the measure of angle is , and the length of is units. If the area, in square units, of triangle can be represented by the expression , where is a constant, what is the value of ?
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Correct Answer: 338The correct answer is . The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. In triangle , it's given that angle is a right angle. Thus, is the leg opposite of angle and is the leg adjacent to angle . It follows that . It's also given that the measure of angle is and the length of is units. Substituting for and for in the equation yields . Multiplying each side of this equation by yields . Therefore, the length of is . The area of a triangle is half the product of the lengths of its legs. Since the length of is and the length of is , it follows that the area of triangle is square units, or square units. It's given that the area, in square units, of triangle can be represented by the expression , where is a constant. Therefore, is the value of .
Question 50 50 of 269 selected Right Triangles & Trigonometry
Triangle above is a right triangle, and
. What is the length of side
?
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The correct answer is 24. The sine of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the sine of angle B, or , is equal to the ratio of the length of side
to the length of side
. It’s given that the length of side
is 26 and that
. Therefore,
. Multiplying both sides of this equation by 26 yields
.
By the Pythagorean Theorem, the relationship between the lengths of the sides of triangle ABC is as follows: , or
. Subtracting 100 from both sides of
yields
. Taking the square root of both sides of
yields
.
Question 51 51 of 269 selected Lines, Angles, & Triangles
In triangles and , angles and each have measure , , and . Which additional piece of information is sufficient to prove that triangle is similar to triangle ?
and
and
The measures of angles and are and , respectively.
The measures of angles and are and , respectively.
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Correct Answer: DChoice D is correct. Two triangles are similar if they have three pairs of congruent corresponding angles. It’s given that angles and each measure , and so these corresponding angles are congruent. If angle is , then angle must be so that the sum of the angles in triangle is . If angle is , then angle must be so that the sum of the angles in triangle is . Therefore, if the measures of angles and are and , respectively, then corresponding angles and are both , and corresponding angles and are both . It follows that triangles and have three pairs of congruent corresponding angles, and so the triangles are similar. Therefore, the additional piece of information that is sufficient to prove that triangle is similar to triangle is that the measures of angles and are and , respectively.
Choice A is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the two sides given are not proportional and the angle given is not included by the given sides.
Choice B is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the angle given is not included between the proportional sides.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 52 52 of 269 selected Area & Volume
The figure shows the lengths, in inches, of two sides of a right triangle. What is the area of the triangle, in square inches?
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Correct Answer: 7.5, 15/2The correct answer is . The area, , of a triangle is given by the formula , where is the length of the base of the triangle and is the height of the triangle. In the right triangle shown, the length of the base of the triangle is inches, and the height is inches. It follows that and . Substituting for and for in the formula yields , which is equivalent to , or . Therefore, the area of the triangle, in square inches, is . Note that 15/2 and 7.5 are examples of ways to enter a correct answer.
Question 53 53 of 269 selected Right Triangles & Trigonometry
In the right triangle shown, what is the value of ?
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Correct Answer: BChoice B is correct. The sine of an acute angle in a right triangle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. The hypotenuse of a right triangle is the side opposite the right angle. In right triangle , side is the side opposite angle and side is the hypotenuse. It's given that the length of side is units and the length of side is units. Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the ratio of the length of the hypotenuse to the length of the side opposite angle rather than the ratio of the length of the side opposite angle to the length of the hypotenuse.
Choice D is incorrect. This is the length of the hypotenuse rather than .
Question 54 54 of 269 selected Right Triangles & Trigonometry
The lengths of the legs of a right triangle are shown. Which of the following is closest to the length of the triangle's hypotenuse?
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Correct Answer: CChoice C is correct. The Pythagorean theorem states that for a right triangle, , where and represent the lengths of the legs of the triangle and represents the length of its hypotenuse. In the triangle shown, the legs have lengths of and . Substituting for and for in the equation yields , which is equivalent to , or . Taking the positive square root of both sides of this equation yields . Thus, the value of is approximately . Therefore, of the given choices, is the closest to the length of the triangle's hypotenuse.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 55 55 of 269 selected Right Triangles & Trigonometry
Which equation shows the relationship between the side lengths of the given triangle?
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Correct Answer: CChoice C is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, , where and are the lengths of the legs and is the length of the hypotenuse. For the given right triangle, the lengths of the legs are and , and the length of the hypotenuse is . Substituting for and for in the equation yields . Thus, the relationship between the side lengths of the given triangle is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 56 56 of 269 selected Circles
The graph of in the xy-plane is a circle. What is the length of the circle’s radius?
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Correct Answer: 10The correct answer is . It's given that the graph of in the xy-plane is a circle. The equation of a circle in the xy-plane can be written in the form , where the coordinates of the center of the circle are and the length of the radius of the circle is . The term in this equation can be obtained by adding the square of half the coefficient of to both sides of the given equation to complete the square. The coefficient of is . Half the coefficient of is . The square of half the coefficient of is . Adding to each side of yields , or . Similarly, the term can be obtained by adding the square of half the coefficient of to both sides of this equation, which yields , or . This equation is equivalent to , or . Therefore, the length of the circle's radius is .
Question 57 57 of 269 selected Right Triangles & Trigonometry
The perimeter of an isosceles right triangle is inches. What is the length, in inches, of the hypotenuse of this triangle?
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Correct Answer: CChoice C is correct. The perimeter of a triangle is the sum of the lengths of its sides. Since the given triangle is an isosceles right triangle, the length of each leg is the same and the length of the hypotenuse is equal to times the length of a leg. Let represent the length, in inches, of a leg of this isosceles right triangle. Therefore, the perimeter, in inches, of the triangle is , or , which is equivalent to . It's given that the perimeter of this triangle is inches. Thus, . Dividing both sides of this equation by yields . Multiplying the right-hand side of this equation by yields , or . It follows that the length, in inches, of a leg of this isosceles right triangle is . Therefore, the length, in inches, of the hypotenuse of this isosceles right triangle is , or .
Choice A is incorrect. If this were the length of the hypotenuse, the perimeter would be inches.
Choice B is incorrect. This is the length, in inches, of a leg of this triangle, not the hypotenuse.
Choice D is incorrect. If this were the length of the hypotenuse, the perimeter would be inches.
Question 58 58 of 269 selected Area & Volume
A cube has an edge length of inches. What is the volume, in cubic inches, of the cube?
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Correct Answer: DChoice D is correct. The volume, , of a cube can be found using the formula , where is the edge length of the cube. It's given that a cube has an edge length of inches. Substituting inches for in this equation yields cubic inches, or cubic inches. Therefore, the volume of the cube is cubic inches.
Choice A is incorrect. This is the perimeter, in inches, of the cube.
Choice B is incorrect. This is the area, in square inches, of a face of the cube.
Choice C is incorrect. This is the surface area, in square inches, of the cube.
Question 59 59 of 269 selected Area & Volume
The length of the edge of the base of a right square prism is units. The volume of the prism is cubic units. What is the height, in units, of the prism?
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Correct Answer: DChoice D is correct. The volume, , of a right square prism is given by the formula , where represents the length of the edge of the base and represents the height of the prism. It’s given that the volume of a right square prism is cubic units and the length of the edge of the base is units. Substituting for and for in the formula yields , or . Dividing both sides of this equation by yields . Therefore, the height, in units, of the prism is .
Choice A is incorrect. This is the height, in units, of a right square prism where the length of the edge of the base is units and the volume of the prism is , not , units.
Choice B is incorrect. This is the area, in square units, of the base, not the height, in units, of the prism.
Choice C is incorrect. This is the height, in units, of a right square prism where the length of the edge of the base is units and the volume of the prism is , not , units.
Question 60 60 of 269 selected Area & Volume
A cylinder has a diameter of inches and a height of inches. What is the volume, in cubic inches, of the cylinder?
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Correct Answer: CChoice C is correct. The base of a cylinder is a circle with a diameter equal to the diameter of the cylinder. The volume, , of a cylinder can be found by multiplying the area of the circular base, , by the height of the cylinder, , or . The area of a circle can be found using the formula , where is the radius of the circle. It’s given that the diameter of the cylinder is inches. Thus, the radius of this circle is inches. Therefore, the area of the circular base of the cylinder is , or square inches. It’s given that the height of the cylinder is inches. Substituting for and for in the formula gives , or cubic inches.
Choice A is incorrect. This is the area of the circular base of the cylinder.
Choice B is incorrect and may result from using , instead of , as the value of in the formula for the area of a circle.
Choice D is incorrect and may result from using , instead of , for the radius of the circular base.
Question 61 61 of 269 selected Right Triangles & Trigonometry
Right triangle is shown. What is the value of ?
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Correct Answer: CChoice C is correct. In the triangle shown, the measure of angle is and angle is a right angle, which means that it has a measure of . Since the sum of the angles in a triangle is equal to , the measure of angle is equal to , or . In a right triangle whose acute angles have measures and , the lengths of the legs can be represented by the expressions , , and , where is the length of the leg opposite the angle with measure , is the length of the leg opposite the angle with measure , and is the length of the hypotenuse. In the triangle shown, the hypotenuse has a length of . It follows that , or . Therefore, the length of the leg opposite angle is and the length of the leg opposite angle is . The tangent of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. The length of the leg opposite angle is and the length of the leg adjacent to angle is . Therefore, the value of is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not the value of .
Choice D is incorrect. This is the length of the leg opposite angle , not the value of .
Question 62 62 of 269 selected Area & Volume
Square A has side lengths that are times the side lengths of square B. The area of square A is times the area of square B. What is the value of ?
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Correct Answer: AChoice A is correct. The area of a square is , where is the side length of the square. Therefore, the area of square B is , where is the side length of square B. It’s given that square A has side lengths that are times the side lengths of square B. Therefore, the side length of square A can be represented by the expression . It follows that the area of square A is , or . It’s given that the area of square A is times the area of square B, so . Therefore, the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 63 63 of 269 selected Area & Volume
A manufacturing company produces two sizes of cylindrical containers that each have a height of 50 centimeters. The radius of container A is 16 centimeters, and the radius of container B is 25% longer than the radius of container A. What is the volume, in cubic centimeters, of container B?
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Correct Answer: BChoice B is correct. If the radius of container A is 16 centimeters and the radius of container B is 25% longer than the radius of container A, then the radius of container B is centimeters. The volume of a cylinder is
, where r is the radius of the cylinder and h is its height. Substituting
and
into
yields that the volume of cylinder B is
cubic centimeters.
Choice A is incorrect and may result from multiplying the radius of cylinder B by the radius of cylinder A rather than squaring the radius of cylinder B. Choice C is incorrect and may result from multiplying the radius of cylinder B by 25 rather than squaring it. Choice D is incorrect and may result from taking the radius of cylinder B to be 25 centimeters rather than 20 centimeters.
Question 64 64 of 269 selected Circles
Point lies on a unit circle in the xy-plane and has coordinates . Point is the center of the circle and has coordinates . Point also lies on the circle and has coordinates , where is a constant. Which of the following could be the positive measure of angle , in radians?
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Correct Answer: DChoice D is correct. It's given that the circle is a unit circle, which means the circle has a radius of . It's also given that point is the center of the circle and has coordinates and that point lies on the circle and has coordinates . Since the radius of the circle is , the value of must be , as all other points with an x-coordinate of are a distance greater than away from point . Since and are points on the unit circle centered at , let side be the starting side of the angle and side be the terminal side of the angle. Since side is on the positive x-axis and side is on the negative x-axis, side is half of a rotation around the unit circle, or radians, away from side . Therefore, the positive measure of angle , in radians, is equal to plus an integer multiple of . In other words, the positive measure of angle , in radians, is an odd integer multiple of . Of the given choices, only is an odd integer multiple of .
Choice A is incorrect. This could be the positive measure of an angle where the starting side is and the terminal side contains the point , not .
Choice B is incorrect. This could be the positive measure of an angle where the starting side is and the terminal side contains the point , not .
Choice C is incorrect. This could be the positive measure of an angle where the starting side is and the terminal side contains the point , not .
Question 65 65 of 269 selected Circles
The graph of the given equation is a circle in the xy-plane. The point lies on the circle. Which of the following is a possible value for ?
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Correct Answer: BChoice B is correct. An equation of the form , where , , and are constants, represents a circle in the xy-plane with center and radius . Therefore, the circle represented by the given equation has center and radius . Since the center of the circle has an x-coordinate of and the radius of the circle is , the least possible x-coordinate for any point on the circle is , or . Similarly, the greatest possible x-coordinate for any point on the circle is , or . Therefore, if the point lies on the circle, it must be true that . Of the given choices, only satisfies this inequality.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 66 66 of 269 selected Area & Volume
Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is , and the surface area of right rectangular prism Y is . The volume of right rectangular prism Y is . What is the sum of the volumes, , of right rectangular prism X and right rectangular prism Y?
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Correct Answer: 1260The correct answer is . Since it's given that prisms X and Y are similar, all the linear measurements of prism Y are times the respective linear measurements of prism X, where is a positive constant. Therefore, the surface area of prism Y is times the surface area of prism X and the volume of prism Y is times the volume of prism X. It's given that the surface area of prism Y is , and the surface area of prism X is , which implies that . Dividing both sides of this equation by yields , or . Since is a positive constant, . It's given that the volume of prism Y is . Therefore, the volume of prism X is equal to , which is equivalent to , or . Thus, the sum of the volumes, in , of the two prisms is , or .
Question 67 67 of 269 selected Circles
The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?
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Correct Answer: BChoice B is correct. The standard equation of a circle in the xy-plane is of the form , where
are the coordinates of the center of the circle and r is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms,
, needs a 100 to complete the square, and the sum of the second two terms,
, needs a 64 to complete the square. Adding 100 and 64 to both sides of the given equation yields
, which is equivalent to
. Therefore, the coordinates of the center of the circle are
.
Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying the coordinates of the center.
Question 68 68 of 269 selected Area & Volume
A rectangle has an area of square meters and a length of meters. What is the width, in meters, of the rectangle?
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Correct Answer: AChoice A is correct. The area , in square meters, of a rectangle is the product of its length , in meters, and its width , in meters; thus, . It's given that a rectangle has an area of square meters and a length of meters. Substituting for and for in the equation yields . Dividing both sides of this equation by yields . Therefore, the width, in meters, of the rectangle is .
Choice B is incorrect. This is the difference between the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.
Choice C is incorrect. This is the square of the length, in meters, not the width, in meters, of the rectangle.
Choice D is incorrect. This is the product of the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.
Question 69 69 of 269 selected Area & Volume
A sphere has a radius of feet. What is the volume, in cubic feet, of the sphere?
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Correct Answer: DChoice D is correct. The volume, , of a sphere can be found using the formula , where is the radius of the sphere. It’s given that the sphere has a radius of feet. Substituting for in the formula yields , which is equivalent to , or . Therefore, the volume, in cubic feet, of the sphere is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the volume, in cubic feet, of a sphere with a radius of feet.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 70 70 of 269 selected Right Triangles & Trigonometry
In the triangle shown, what is the value of ?
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Correct Answer: .6956, .6957, 16/23The correct answer is . In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the length of the side opposite the angle with measure is units and the length of the hypotenuse is units. Therefore, the value of is . Note that 16/23, .6956, .6957, 0.695, and 0.696 are examples of ways to enter a correct answer.
Question 71 71 of 269 selected Circles
Point O is the center of the circle above, and the measure of is
. If the length of
is 18, what is the length of arc
?
Show Answer
Correct Answer: BChoice B is correct. Because segments OA and OB are radii of the circle centered at point O, these segments have equal lengths. Therefore, triangle AOB is an isosceles triangle, where angles OAB and OBA are congruent base angles of the triangle. It’s given that angle OAB measures . Therefore, angle OBA also measures
. Let
represent the measure of angle AOB. Since the sum of the measures of the three angles of any triangle is
, it follows that
, or
. Subtracting
from both sides of this equation yields
, or
radians. Therefore, the measure of angle AOB, and thus the measure of arc
, is
radians. Since
is a radius of the given circle and its length is 18, the length of the radius of the circle is 18. Therefore, the length of arc
can be calculated as
, or
.
Choices A, C, and D are incorrect and may result from conceptual or computational errors.
Question 72 72 of 269 selected Right Triangles & Trigonometry
In , is a right angle and the length of is millimeters. If , what is the length, in millimeters, of ?
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Correct Answer: BChoice B is correct. It's given that in , is a right angle. Therefore, is a right triangle, and is the hypotenuse of the triangle. It's also given that . Since the cosine of an acute angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, the ratio of the length of to the length of is to . It follows that the length of can be represented as and the length of can be represented as , where is a constant. The Pythagorean theorem states that the sum of the squares of the length of the legs of a right triangle is equal to the square of the length of its hypotenuse, so it follows that . Substituting for and for in this equation yields , or . Subtracting from both sides of this equation yields , or . It follows that the ratio of the length of to the length of is to . Let represent the length, in millimeters, of . It's given that the length of is millimeters. Since the ratio of the length of to the length of is to , . Multiplying both sides of this equation by yields , or . Therefore, the length of is millimeters.
Choice A is incorrect. This is the scale factor by which the to to ratio is multiplied that results in the side lengths of .
Choice C is incorrect. This is the length of , not the length of .
Choice D is incorrect. This is the length of , not the length of .
Question 73 73 of 269 selected Circles
A circle in the xy-plane has a diameter with endpoints and . An equation of this circle is , where is a positive constant. What is the value of ?
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Correct Answer: 5The correct answer is . The standard form of an equation of a circle in the xy-plane is , where , , and are constants, the coordinates of the center of the circle are , and the length of the radius of the circle is . It′s given that an equation of the circle is . Therefore, the center of this circle is . It’s given that the endpoints of a diameter of the circle are and . The length of the radius is the distance from the center of the circle to an endpoint of a diameter of the circle, which can be found using the distance formula, . Substituting the center of the circle and one endpoint of the diameter in this formula gives a distance of , or , which is equivalent to . Since the distance from the center of the circle to an endpoint of a diameter is , the value of is .
Question 74 74 of 269 selected Lines, Angles, & Triangles
Each side of equilateral triangle S is multiplied by a scale factor of to create equilateral triangle T. The length of each side of triangle T is greater than the length of each side of triangle S. Which of the following could be the value of ?
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Correct Answer: AChoice A is correct. It's given that each side of equilateral triangle S is multiplied by a scale factor of to create equilateral triangle T. Since the length of each side of triangle T is greater than the length of each side of triangle S, the scale factor of must be greater than . Of the given choices, only is greater than .
Choice B is incorrect. If each side of equilateral triangle S is multiplied by a scale factor of , the length of each side of triangle T would be equal to the length of each side of triangle S.
Choice C is incorrect. If each side of equilateral triangle S is multiplied by a scale factor of , the length of each side of triangle T would be less than the length of each side of triangle S.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 75 75 of 269 selected Area & Volume
A right circular cylinder has a height of and a base with a radius of . What is the volume, , of the cylinder?
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Correct Answer: DChoice D is correct. The volume, , of a right circular cylinder is given by , where is the radius of the circular base and is the height of the cylinder. It’s given that the cylinder has a height of meters and a base with a radius of meters. Substituting for and for in yields , or . Therefore, the volume, in , of the cylinder is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the volume, in , of a cylinder with a radius of meters and a height of meters.
Question 76 76 of 269 selected Area & Volume
Rectangle is similar to rectangle . The area of rectangle is square inches, and the area of rectangle is square inches. The length of the longest side of rectangle is inches. What is the length, in inches, of the longest side of rectangle ?
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Correct Answer: CChoice C is correct. It's given that rectangle is similar to rectangle . Therefore, if the length of each side of rectangle is times the length of the corresponding side of rectangle , then the area of rectangle is times the area of rectangle . It’s given that the area of rectangle is square inches and the area of rectangle is square inches. It follows that , or . Taking the square root of each side of this equation yields , or . It follows that the length of each side of rectangle is times the length of the corresponding side of rectangle . It’s given that the length of the longest side of rectangle is inches. Therefore, inches is times the length of the longest side of rectangle , and the longest side of rectangle is equal to , or , inches.
Choice A is incorrect. This is the length, in inches, of the longest side of a rectangle with side lengths that are the corresponding side lengths of rectangle , rather than a rectangle with an area that is the area of rectangle .
Choice B is incorrect. This is the factor by which the area of rectangle is larger than the area of rectangle , not the length, in inches, of the longest side of rectangle .
Choice D is incorrect. This is the length, in inches, of the longest side of rectangle , not rectangle .
Question 77 77 of 269 selected Right Triangles & Trigonometry
A graphic designer is creating a logo for a company. The logo is shown in the figure above. The logo is in the shape of a trapezoid and consists of three congruent equilateral triangles. If the perimeter of the logo is 20 centimeters, what is the combined area of the shaded regions, in square centimeters, of the logo?
16
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Correct Answer: CChoice C is correct. It’s given that the logo is in the shape of a trapezoid that consists of three congruent equilateral triangles, and that the perimeter of the trapezoid is 20 centimeters (cm). Since the perimeter of the trapezoid is the sum of the lengths of 5 of the sides of the triangles, the length of each side of an equilateral triangle is . Dividing up one equilateral triangle into two right triangles yields a pair of congruent 30°-60°-90° triangles. The shorter leg of each right triangle is half the length of the side of an equilateral triangle, or 2 cm. Using the Pythagorean Theorem,
, the height of the equilateral triangle can be found. Substituting
and
and solving for b yields
cm. The area of one equilateral triangle is
, where
and
. Therefore, the area of one equilateral triangle is
. The shaded area consists of two such triangles, so its area is
.
Alternate approach: The area of a trapezoid can be found by evaluating the expression , where
is the length of one base,
is the length of the other base, and h is the height of the trapezoid. Substituting
,
, and
yields the expression
, or
, which gives an area of
for the trapezoid. Since two-thirds of the trapezoid is shaded, the area of the shaded region is
.
Choice A is incorrect. This is the height of the trapezoid. Choice B is incorrect. This is the area of one of the equilateral triangles, not two. Choice D is incorrect and may result from using a height of 4 for each triangle rather than the height of .
Question 78 78 of 269 selected Circles
Circle A in the xy-plane has the equation . Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the xy-plane is , where is a constant. What is the value of ?
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Correct Answer: 16The correct answer is . An equation of a circle in the xy-plane can be written as , where the center of the circle is , the radius of the circle is , and where , , and are constants. It’s given that the equation of circle A is , which is equivalent to . Therefore, the center of circle A is and the radius of circle A is . It’s given that circle B has the same center as circle A and that the radius of circle B is two times the radius of circle A. Therefore, the center of circle B is and the radius of circle B is , or . Substituting for , for , and for into the equation yields , which is equivalent to . It follows that the equation of circle B in the xy-plane is . Therefore, the value of is .
Question 79 79 of 269 selected Circles
What is the diameter of the circle in the xy-plane with equation ?
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Correct Answer: BChoice B is correct. The standard form of an equation of a circle in the xy-plane is , where the coordinates of the center of the circle are and the length of the radius of the circle is . For the circle in the xy-plane with equation , it follows that . Taking the square root of both sides of this equation yields or . Because represents the length of the radius of the circle and this length must be positive, . Therefore, the radius of the circle is . The diameter of a circle is twice the length of the radius of the circle. Thus, yields . Therefore, the diameter of the circle is .
Choice A is incorrect. This is the radius of the circle.
Choice C is incorrect. This is the square of the radius of the circle.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 80 80 of 269 selected Lines, Angles, & Triangles
Triangles ABC and DEF are shown above. Which of the following is equal to the ratio ?
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Correct Answer: BChoice B is correct. In right triangle ABC, the measure of angle B must be 58° because the sum of the measure of angle A, which is 32°, and the measure of angle B is 90°. Angle D in the right triangle DEF has measure 58°. Hence, triangles ABC and DEF are similar (by angle-angle similarity). Since is the side opposite to the angle with measure 32° and AB is the hypotenuse in right triangle ABC, the ratio
is equal to
.
Alternate approach: The trigonometric ratios can be used to answer this question. In right triangle ABC, the ratio . The angle E in triangle DEF has measure 32° because
. In triangle DEF, the ratio
. Therefore,
.
Choice A is incorrect because is the reciprocal of the ratio
. Choice C is incorrect because
, not
. Choice D is incorrect because
, not
.
Question 81 81 of 269 selected Right Triangles & Trigonometry
In the triangle shown, what is the value of ?
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Correct Answer: CChoice C is correct. The tangent of an acute angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the shorter side adjacent to the angle. In the triangle shown, the length of the side opposite the angle with measure is units and the length of the side adjacent to the angle with measure is units. Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 82 82 of 269 selected Lines, Angles, & Triangles
In the figure shown, triangle is similar to triangle . The measure of angle is , and . What is the measure of angle ?
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Correct Answer: CChoice C is correct. It's given that triangle is similar to triangle . Corresponding angles in similar triangles have equal measure. Angle and angle represent the same angle. It follows that angle and angle have equal measure and are corresponding angles. It's given in the figure that angle and angle are right angles and therefore have equal measure. It follows that angle and angle are corresponding angles. Therefore, angle and angle are corresponding angles and have equal measure. It's given that the measure of angle is , so the measure of angle is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Question 83 83 of 269 selected Area & Volume
The three points shown define a circle. The circumference of this circle is , where is a constant. What is the value of ?
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Correct Answer: 8The correct answer is . It's given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point is halfway between the points and , and is a distance of units from each of those two points. The point is also a distance of units from . Because the point is the same distance from all three points shown, it must be the center of the circle. Since that distance is , it follows that the radius of the circle is . The circumference of a circle with radius is equal to . It follows that the circumference of the given circle is , or . It's given that the circumference of the circle is . Therefore, the value of is .
Question 84 84 of 269 selected Area & Volume
The floor of a ballroom has an area of square meters. An architect creates a scale model of the floor of the ballroom, where the length of each side of the model is times the length of the corresponding side of the actual floor of the ballroom. What is the area, in square meters, of the scale model?
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Correct Answer: AChoice A is correct. It’s given that the length of each side of a scale model is times the length of the corresponding side of the actual floor of a ballroom. Therefore, the area of the scale model is , or , times the area of the actual floor of the ballroom. It’s given that the area of the floor of the ballroom is square meters. Therefore, the area, in square meters, of the scale model is , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 85 85 of 269 selected Area & Volume
A cube has a volume of cubic units. What is the surface area, in square units, of the cube?
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Correct Answer: 36504The correct answer is . The volume of a cube can be found using the formula , where represents the edge length of a cube. It’s given that this cube has a volume of cubic units. Substituting for in yields . Taking the cube root of both sides of this equation yields . Thus, the edge length of the cube is units. Since each face of a cube is a square, it follows that each face has an edge length of units. The area of a square can be found using the formula . Substituting for in this formula yields , or . Therefore, the area of one face of this cube is square units. Since a cube has faces, the surface area, in square units, of this cube is , or .
Question 86 86 of 269 selected Lines, Angles, & Triangles
In the figure shown, line intersects lines and . Which additional piece of information is sufficient to prove that lines and are parallel?
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Correct Answer: AChoice A is correct. In the figure shown, lines and are parallel if and only if a pair of corresponding angles are congruent. It's given that one angle has a measure of and that the corresponding angle has a measure of . Therefore, is sufficient to prove that lines and are parallel.
Choice B is incorrect. The angle measuring and the angle measuring are alternate interior angles. Thus, if lines and are parallel, is equal to , not .
Choice C is incorrect. The angle measuring and the angle measuring are vertical angles. Thus, , whether lines and are parallel or not.
Choice D is incorrect. The angle measuring is supplementary to the angle measuring . Thus, , or , whether lines and are parallel or not.
Question 87 87 of 269 selected Lines, Angles, & Triangles
In right triangle , angle is the right angle and . Point on side is connected by a line segment with point on side such that line segment is parallel to side and . What is the length of line segment ?
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Correct Answer: 54The correct answer is . It’s given that in triangle , point on side is connected by a line segment with point on side such that line segment is parallel to side . It follows that parallel segments and are intersected by sides and . If two parallel segments are intersected by a third segment, corresponding angles are congruent. Thus, corresponding angles and are congruent and corresponding angles and are congruent. Since triangle has two angles that are each congruent to an angle in triangle , triangle is similar to triangle by the angle-angle similarity postulate, where side corresponds to side , and side corresponds to side . Since the lengths of corresponding sides in similar triangles are proportional, it follows that . Since point lies on side , . It's given that . Substituting for in the equation yields , or . It’s given that . Substituting for and for in the equation yields , or . Multiplying both sides of this equation by yields . Thus, the length of line segment is .
Question 88 88 of 269 selected Area & Volume
The perimeter of triangle is inches, the length of side is inches, and the length of side is inches. What is the length, in inches, of side ?
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Correct Answer: BChoice B is correct. The perimeter of a triangle is the sum of the lengths of all three sides of the triangle. It’s given that the lengths of side and side are inches and inches, respectively. Let represent the length, in inches, of side . The sum of the lengths, in inches, of all three sides of triangle can be represented by the expression . Since it’s given that the perimeter of triangle is inches, it follows that , or . Subtracting from both sides of this equation yields . Therefore, the length, in inches, of side is .
Choice A is incorrect. This is the length, in inches, of side .
Choice C is incorrect. This is the length, in inches, of side .
Choice D is incorrect. This is the sum of the lengths, in inches, of sides and .
Question 89 89 of 269 selected Area & Volume
The table gives the perimeters of similar triangles and , where corresponds to . The length of is .
| Perimeter | |
|---|---|
| Triangle | |
| Triangle |
What is the length of ?
Show Answer
Correct Answer: DChoice D is correct. It's given that triangle is similar to triangle . Therefore, each side of triangle is times its corresponding side of triangle , where is a constant. It follows that the perimeter of triangle is times the perimeter of triangle . It's also given that corresponds to and the length of is . Let represent the length of . It follows that . The table shows that the perimeters of triangles and are and , respectively. It follows that , or . Substituting for in the equation yields , or . Therefore, the length of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the length of , not the length of .
Choice C is incorrect and may result from conceptual or calculation errors.
Question 90 90 of 269 selected Right Triangles & Trigonometry
The perimeter of an equilateral triangle is centimeters. The height of this triangle is centimeters, where is a constant. What is the value of ?
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Correct Answer: 104The correct answer is . An equilateral triangle is a triangle in which all three sides have the same length and all three angles have a measure of . The height of the triangle, , is the length of the altitude from one vertex. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the angle in each 30-60-90 right triangle. Since the altitude has a length of , it follows from the properties of 30-60-90 right triangles that the side across from each angle has a length of and each hypotenuse has a length of . In this case, the hypotenuse of each 30-60-90 right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral triangle is . The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral triangle is ; therefore, , or . Dividing both sides of this equation by yields .
Question 91 91 of 269 selected Area & Volume
A right circular cone has a height of and a base with a diameter of . The volume of this cone is . What is the value of ?
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Correct Answer: 66The correct answer is . It’s given that the right circular cone has a height of centimeters and a base with a diameter of . Since the diameter of the base of the cone is , the radius of the base is . The volume , , of a right circular cone can be found using the formula , where is the height, , and is the radius, , of the base of the cone. Substituting for and for in this formula yields , or . Therefore, the volume of the cone is . It’s given that the volume of the cone is . Therefore, the value of is .
Question 92 92 of 269 selected Right Triangles & Trigonometry
In triangle , angle is a right angle. The length of side is and the length of side is . What is the length of side ?
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Correct Answer: BChoice B is correct. The Pythagorean theorem states that for a right triangle, , where represents the length of the hypotenuse and and represent the lengths of the legs. It’s given that in triangle , angle is a right angle. Therefore, triangle is a right triangle, where the hypotenuse is side and the legs are sides and . It’s given that the lengths of sides and are and , respectively. Substituting these values for and in the formula yields , which is equivalent to , or . Taking the square root of both sides of this equation yields . Since represents the length of the hypotenuse, side , must be positive. Therefore, the length of side is .
Choice A is incorrect. This is the result of solving the equation , not .
Choice C is incorrect. This is the result of solving the equation , not .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 93 93 of 269 selected Lines, Angles, & Triangles
In the figure above, what is the length of ?
2.2
2.3
2.4
2.5
Show Answer
Correct Answer: CChoice C is correct. First, is the hypotenuse of right
, whose legs have lengths 3 and 4. Therefore,
, so
and
. Second, because
corresponds to
and because
corresponds to
,
is similar to
. The ratio of corresponding sides of similar triangles is constant, so
. Since
and it’s given that
and
,
. Solving for NQ results in
, or 2.4.
Choices A, B, and D are incorrect and may result from setting up incorrect ratios.
Question 94 94 of 269 selected Area & Volume
A rectangle has a length of units and a width of units. Which expression gives the area, in square units, of this rectangle?
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Correct Answer: DChoice D is correct. The area of a rectangle is given by , where is the length of the rectangle and is the width of the rectangle. It's given that a rectangle has a length of units and a width of units. It follows that the area of the rectangle is square units. Therefore, the expression that gives the area, in square units, of this rectangle, is .
Choice A is incorrect. This expression gives the perimeter, in units, of this rectangle.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Question 95 95 of 269 selected Lines, Angles, & Triangles
Triangle is similar to triangle such that , , and correspond to , , and , respectively. The measure of is and . What is the measure of ?
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Correct Answer: CChoice C is correct. It’s given that triangle is similar to triangle , such that , , and correspond to , , and , respectively. Since corresponding angles of similar triangles are congruent, it follows that the measure of is congruent to the measure of . It’s given that the measure of is . Therefore, the measure of is .
Choice A is incorrect and may result from a conceptual error.
Choice B is incorrect. This is half the measure of .
Choice D is incorrect. This is twice the measure of .
Question 96 96 of 269 selected Circles
Point is the center of a circle. The measure of arc on this circle is . What is the measure, in degrees, of its associated angle ?
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Correct Answer: 100The correct answer is . It's given that point is the center of a circle and the measure of arc on the circle is . It follows that points and lie on the circle. Therefore, and are radii of the circle. A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. Therefore, is a central angle. Because the degree measure of an arc is equal to the measure of its associated central angle, it follows that the measure, in degrees, of is .
Question 97 97 of 269 selected Area & Volume
The table gives the perimeters of similar triangles and , where corresponds to . The length of is .
| Perimeter | |
|---|---|
| Triangle | |
| Triangle |
What is the length of ?
Show Answer
Correct Answer: CChoice C is correct. It’s given that triangle is similar to triangle , and corresponds to . If two triangles are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides. It’s given that the perimeter of triangle is , the perimeter of triangle is , and the length of is . Let represent the length of . It follows that , or . Multiplying each side of this equation by yields . Multiplying each side of this equation by yields . Therefore, the length of is .
Choice A is incorrect. This is the solution to , not .
Choice B is incorrect. This is the length of , not .
Choice D is incorrect. This is the sum of the length of and the perimeter of triangle , not the length of .
Question 98 98 of 269 selected Lines, Angles, & Triangles
In the figure shown, lines and are parallel, and line intersects both lines. If , which of the following must be true?
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Correct Answer: BChoice B is correct. In the figure shown, the angle measuring is congruent to its vertical angle formed by lines and , so the measure of the vertical angle is also . The vertical angle forms a same-side interior angle pair with the angle measuring . It's given that lines and are parallel. Therefore, same-side interior angles in the figure are supplementary, which means the sum of the measure of the vertical angle and the measure of the angle measuring is , or . Subtracting from both sides of this equation yields . Substituting for in the inequality yields . Adding to both sides of this inequality yields . Subtracting from both sides of this inequality yields , or . Thus, if , it must be true that .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. must be equal to, not less than, .
Choice D is incorrect. must be equal to, not greater than, .
Question 99 99 of 269 selected Area & Volume
A circle has a circumference of centimeters. What is the diameter, in centimeters, of the circle?
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Correct Answer: 31The correct answer is . The circumference of a circle is equal to centimeters, where represents the radius, in centimeters, of the circle, and the diameter of the circle is equal to centimeters. It's given that a circle has a circumference of centimeters. Therefore, . Dividing both sides of this equation by yields . Since the diameter of the circle is equal to centimeters, it follows that the diameter, in centimeters, of the circle is .
Question 100 100 of 269 selected Circles
The equation defines a circle in the xy‑plane. What is the radius of the circle?
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The correct answer is 11. A circle with equation , where a, b, and r are constants, has center
and radius r. Therefore, the radius of the given circle is
, or 11.
Question 101 101 of 269 selected Area & Volume
Circle has a radius of . Circle has an area of . What is the total area, , of circles and ?
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Correct Answer: DChoice D is correct. The area, , of a circle is given by the formula , where represents the radius of the circle. It’s given that circle has a radius of . Substituting for in the formula yields , or . Therefore, the area of circle is . It’s given that circle has an area of . Therefore, the total area, , of circles and is , or .
Choice A is incorrect. This is the sum of the radii, , of circles and multiplied by , not the total area, , of the circles.
Choice B is incorrect. This is the sum of the diameters, , of circles and multiplied by , not the total area, , of the circles.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 102 102 of 269 selected Circles
A circle in the xy-plane has the equation . Which of the following gives the center of the circle and its radius?
The center is at and the radius is .
The center is at and the radius is .
The center is at and the radius is .
The center is at and the radius is .
Show Answer
Correct Answer: AChoice A is correct. For a circle in the xy-plane that has the equation , where , , and are constants, is the center of the circle and the positive value of is the radius of the circle. In the given equation, and . Taking the square root of each side of yields . Therefore, the center of the circle is at and the radius is .
Choice B is incorrect. This gives the center and radius of a circle with equation , not .
Choice C is incorrect. This gives the center and radius of a circle with equation , not .
Choice D is incorrect. This gives the center and radius of a circle with equation , not .
Question 103 103 of 269 selected Lines, Angles, & Triangles
Triangle is similar to triangle , such that , , and correspond to , , and respectively. The length of each side of triangle is times the length of its corresponding side in triangle . The measure of side is . What is the measure of side ?
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Correct Answer: DChoice D is correct. It's given that triangle is similar to triangle , such that , , and correspond to , , and , respectively. Therefore, side corresponds to side . Since the length of each side of triangle is times the length of its corresponding side in triangle , it follows that the measure of side is times the measure of side . Thus, since the measure of side is , then the measure of side is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the measure of side , not side .
Choice C is incorrect and may result from conceptual or calculation errors.
Question 104 104 of 269 selected Lines, Angles, & Triangles
In , the measure of is and the measure of is . What is the measure of ?
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Correct Answer: CChoice C is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is . It's given that in , the measure of is and the measure of is . It follows that the measure of is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the sum of the measures of and , not the measure of .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 105 105 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line , and line intersects both lines. What is the value of ?
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Correct Answer: DChoice D is correct. It’s given that line is parallel to line , and line intersects both lines. It follows that line is a transversal. When two lines are parallel and intersected by a transversal, exterior angles on the same side of the transversal are supplementary. Thus, . Subtracting from both sides of this equation yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 106 106 of 269 selected Lines, Angles, & Triangles
In triangle , the measures of and are each . What is the measure of , in degrees? (Disregard the degree symbol when entering your answer.)
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Correct Answer: 84The correct answer is . The sum of the measures of the interior angles of a triangle is . It's given that in triangle , the measures of and are each . Adding the measures, in degrees, of and gives , or . Therefore, the measure of , in degrees, is , or .
Question 107 107 of 269 selected Lines, Angles, & Triangles
In , the measure of is . Which of the following could be the measure, in degrees, of ?
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Correct Answer: AChoice A is correct. The sum of the measures of the angles of a triangle is . Therefore, the sum of the measures of , , and is . It's given that the measure of is . It follows that the sum of the measures of and is , or . Therefore, the measure of , in degrees, must be less than . Of the given choices, only is less than . Thus, the measure, in degrees, of could be .
Choice B is incorrect. If the measure of is , then the sum of the measures of the angles of the triangle is greater than, not equal to, .
Choice C is incorrect. If the measure of is , then the sum of the measures of the angles of the triangle is greater than, not equal to, .
Choice D is incorrect. This is the sum of the measures of the angles of a triangle, in degrees.
Question 108 108 of 269 selected Circles
A circle in the xy-plane has its center at . Line is tangent to this circle at the point . What is the slope of line ?
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Correct Answer: AChoice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has its center at and line is tangent to the circle at the point . The slope of a radius defined by the points and can be calculated as . The points and define the radius of the circle at the point of tangency. Therefore, the slope of this radius can be calculated as , or . If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the slope of the radius. The negative reciprocal of is . Thus, the slope of line is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 109 109 of 269 selected Area & Volume
A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. Which of the following expresses the volume, V, in cubic inches, of such containers, where r is the radius, in inches?
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Correct Answer: DChoice D is correct. The volume, V, of a right cylinder is given by the formula , where r represents the radius of the base of the cylinder and h represents the height. Since the height is 4 inches longer than the radius, the expression
represents the height of each cylindrical container. It follows that the volume of each container is represented by the equation
. Distributing the expression
into each term in the parentheses yields
.
Choice A is incorrect and may result from representing the height as instead of
. Choice B is incorrect and may result from representing the height as
instead of
. Choice C is incorrect and may result from representing the volume of a right cylinder as
instead of
.
Question 110 110 of 269 selected Lines, Angles, & Triangles
In triangle , the measure of angle is and the measure of angle is . In triangle , the measure of angle is and the measure of angle is . Which of the following additional pieces of information is needed to determine whether triangle is similar to triangle ?
The measure of angle
The measure of angle
The measure of angle and the measure of angle
No additional information is needed.
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Correct Answer: DChoice D is correct. When two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. It's given that in triangle , the measure of angle is and the measure of angle is . It's also given that in triangle , the measure of angle is and the measure of angle is . It follows that angle is congruent to angle and that angle is congruent to angle . Therefore, triangle is similar to triangle and no additional information is needed.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Question 111 111 of 269 selected Area & Volume
Rectangle P has an area of square inches. If a rectangle with an area of square inches is removed from rectangle P, what is the area, in square inches, of the resulting figure?
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Correct Answer: DChoice D is correct. It's given that rectangle P has an area of square inches. If a rectangle with an area of square inches is removed from rectangle P, the area, in square inches, of the resulting figure is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 112 112 of 269 selected Circles
Circle A has equation . In the -plane, circle B is obtained by translating circle A to the right units. Which equation represents circle B?
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Correct Answer: CChoice C is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is and the radius of the circle is units. It’s given that circle A has the equation , which can be written as . It follows that , , and . Therefore, the center of circle A is and its radius is unit. If circle A is translated units to the right, the x-coordinate of the center will increase by , while the y-coordinate and the radius of the circle will remain unchanged. Translating the center of circle A to the right units yields , or . Therefore, the center of circle B is . Substituting for , for , and for into the equation yields , or . Therefore, the equation represents circle B.
Choice A is incorrect. This equation represents a circle obtained by shifting circle A down, rather than right, units.
Choice B is incorrect. This equation represents a circle obtained by shifting circle A left, rather than right, units.
Choice D is incorrect. This equation represents a circle obtained by shifting circle A up, rather than right, units.
Question 113 113 of 269 selected Right Triangles & Trigonometry
Triangle is similar to triangle , where angle corresponds to angle , and angles and are right angles. If , what is the value of ?
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Correct Answer: AChoice A is correct. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to that angle to the length of the hypotenuse. It's given that angle is a right angle in triangle and that angle F is a right angle in triangle . Therefore, is equal to the ratio of the length of side to the length of side , and is equal to the ratio of the length of side to the length of side . It’s also given that triangle is similar to triangle , where angle corresponds to angle . Since similar triangles have proportional side lengths, . Therefore, the value of is equal to the value of . Since , the value of is .
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Question 114 114 of 269 selected Right Triangles & Trigonometry
In the figure above, . If
and
, what is the length of
?
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The correct answer is 6. Since ,
and
are both similar to 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since
, which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of
to the 3-4-5 triangle is 3:1. Therefore, the length of
(the side opposite to
) is
, and the length of
(the side adjacent to
) is
. It is also given that
. Since
and
, it follows that
, which means that the similarity ratio of
to the 3-4-5 triangle is 2:1 (
is the side adjacent to
). Therefore, the length of
, which is the side opposite to
, is
.
Question 115 115 of 269 selected Lines, Angles, & Triangles
Quadrilateral is similar to quadrilateral , where , , , and correspond to , , , and , respectively. The measure of angle is , the measure of angle is , and the measure of angle is . The length of each side of is times the length of each corresponding side of . What is the measure of angle ?
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Correct Answer: BChoice B is correct. It's given that quadrilateral is similar to quadrilateral , where , , , and correspond to , , , and , respectively. Since corresponding angles of similar quadrilaterals are congruent, it follows that the measure of angle is equal to the measure of angle . It's given that the measure of angle is . Therefore, the measure of angle is .
Choice A is incorrect. This is the measure of angle .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is times the measure of angle .
Question 116 116 of 269 selected Lines, Angles, & Triangles
A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of these eight angles is . The sum of the measures of four of the eight angles is . Which of the following could NOT be equivalent to , for all values of ?
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Correct Answer: AChoice A is correct. It’s given that a line intersects two parallel lines, forming four acute angles and four obtuse angles. Since there are two parallel lines intersected by a transversal, all four acute angles have the same measure and all four obtuse angles have the same measure. Additionally, each acute angle is supplementary to each obtuse angle. It’s given that the measure of one of these eight angles is . It follows that a supplementary angle has measure , or . It’s also given that the sum of the measures of four of the eight angles is . It follows that the possible values of are ; ; ; ; and . These values are equivalent to ; ; ; ; and , respectively. It follows that of the given choices, only could NOT be equivalent to , for all values of .
Choice B is incorrect. This is the sum of three angles with measure and one angle with measure .
Choice C is incorrect. This is the sum of four angles with measure .
Choice D is incorrect. This is the sum of two angles with measure and two angles with measure .
Question 117 117 of 269 selected Circles
The number of radians in a 720-degree angle can be written as , where a is a constant. What is the value of a ?
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The correct answer is 4. There are radians in a
angle. An angle measure of
is 4 times greater than an angle measure of
. Therefore, the number of radians in a
angle is
.
Question 118 118 of 269 selected Lines, Angles, & Triangles
In the right triangle shown, what is the value of ?
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Correct Answer: BChoice B is correct. The triangle shown is a right triangle, where the interior angle shown with a right angle symbol has a measure of . It's shown that the other two interior angles measure and . The sum of the measures of the interior angles of a triangle is ; therefore, . Combining like terms on the left-hand side of this equation yields . Subtracting from both sides of this equation yields .
Choice A is incorrect. This is the measure, in degrees, of the other acute interior angle of the right triangle, not the value of .
Choice C is incorrect. This is the measure, in degrees, of the right angle of the right triangle, not the value of .
Choice D is incorrect. This is the sum of the measures, in degrees, of the other two interior angles of the right triangle, not the value of .
Question 119 119 of 269 selected Lines, Angles, & Triangles
At a certain time and day, the Washington Monument in Washington, DC, casts a shadow that is 300 feet long. At the same time, a nearby cherry tree casts a shadow that is 16 feet long. Given that the Washington Monument is approximately 555 feet tall, which of the following is closest to the height, in feet, of the cherry tree?
10
20
30
35
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Correct Answer: CChoice C is correct. There is a proportional relationship between the height of an object and the length of its shadow. Let c represent the height, in feet, of the cherry tree. The given relationship can be expressed by the proportion . Multiplying both sides of this equation by 16 yields
. This height is closest to the value given in choice C, 30.
Choices A, B, and D are incorrect and may result from calculation errors.
Question 120 120 of 269 selected Lines, Angles, & Triangles
In the figure above, intersects
at C. If
, what is the value of y ?
100
90
80
60
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Correct Answer: CChoice C is correct. It’s given that ; therefore, substituting 100 for x in triangle ABC gives two known angle measures for this triangle. The sum of the measures of the interior angles of any triangle equals 180°. Subtracting the two known angle measures of triangle ABC from 180° gives the third angle measure:
. This is the measure of angle BCA. Since vertical angles are congruent, the measure of angle DCE is also 60°. Subtracting the two known angle measures of triangle CDE from 180° gives the third angle measure:
. Therefore, the value of y is 80.
Choice A is incorrect and may result from a calculation error. Choice B is incorrect and may result from classifying angle CDE as a right angle. Choice D is incorrect and may result from finding the measure of angle BCA or DCE instead of the measure of angle CDE.
Question 121 121 of 269 selected Right Triangles & Trigonometry
For the triangle shown, which expression represents the value of ?
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Correct Answer: AChoice A is correct. For the right triangle shown, the lengths of the legs are units and units, and the length of the hypotenuse is units. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, . Subtracting from both sides of this equation yields . Taking the square root of both sides of this equation yields . Since is a length, must be positive. Therefore, . Thus, for the triangle shown, represents the value of .
Choice B is incorrect. For the triangle shown, this expression represents the value of , not .
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Question 122 122 of 269 selected Circles
What is the value of ?
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Correct Answer: CChoice C is correct. The cosine of an angle is equal to the cosine of radians more than the angle, where is an integer constant. Since is equivalent to , can be rewritten as , which is equal to . Therefore, the value of is equal to the value of , which is .
Alternate approach: A trigonometric ratio can be found using the unit circle, that is, a circle with radius unit. The cosine of a number is the x-coordinate of the point resulting from traveling a distance of counterclockwise from the point around a unit circle centered at the origin in the xy-plane. A unit circle has a circumference of . It follows that since is equal to , traveling a distance of counterclockwise around a unit circle means traveling around the circle completely times and then another beyond that. That is, traveling results in the same point as traveling . Traveling counterclockwise from the point around a unit circle centered at the origin in the xy-plane results in the point . Thus, the value of is the x-coordinate of the point , which is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of the cosine of a multiple of , not .
Choice D is incorrect. This is the value of , not .
Question 123 123 of 269 selected Circles
Points , , and lie on the circle shown. On this circle, minor arc has a length of centimeters and major arc has a length of centimeters. What is the circumference, in centimeters, of the circle shown?
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Correct Answer: DChoice D is correct. Since the endpoints of minor arc and major arc are the same, and the arcs together form a full circle, the sum of the lengths of these two arcs is equal to the circumference of the circle. It's given that the length of minor arc is centimeters and the length of major arc is centimeters. Therefore, the circumference of the circle, in centimeters, is , or .
Choice A is incorrect. This is the length, in centimeters, of minor arc , not the circumference, in centimeters, of the circle.
Choice B is incorrect. This is the difference of the lengths of major arc and minor arc , in centimeters.
Choice C is incorrect. This is the length, in centimeters, of major arc , not the circumference, in centimeters, of the circle.
Question 124 124 of 269 selected Lines, Angles, & Triangles
In triangle ABC, the measure of angle A is . If triangle ABC is isosceles, which of the following is NOT a possible measure of angle B ?
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Correct Answer: DChoice D is correct. The sum of the three interior angles in a triangle is . It’s given that angle A measures
. If angle B measured
, the measure of angle C would be
. Thus, the measures of the angles in the triangle would be
,
, and
. However, an isosceles triangle has two angles of equal measure. Therefore, angle B can’t measure
.
Choice A is incorrect. If angle B has measure , then angle C would measure
, and
,
, and
could be the angle measures of an isosceles triangle. Choice B is incorrect. If angle B has measure
, then angle C would measure
, and
,
, and
could be the angle measures of an isosceles triangle. Choice C is incorrect. If angle B has measure
, then angle C would measure
, and
,
, and
could be the angle measures of an isosceles triangle.
Question 125 125 of 269 selected Lines, Angles, & Triangles
In the figure above, . What is the value of x ?
72
66
64
58
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Correct Answer: CChoice C is correct. Since , it follows that
is an isosceles triangle with base RU. Therefore,
and
are the base angles of an isosceles triangle and are congruent. Let the measures of both
and
be
. According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is
. Therefore,
, so
.
Note that is the same angle as
. Thus, the measure of
is
. According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore,
is equal to the sum of the measures of
and
; that is,
. Thus, the value of x is 64.
Choice B is incorrect. This is the measure of , but
is not congruent to
. Choices A and D are incorrect and may result from a calculation error.
Question 126 126 of 269 selected Lines, Angles, & Triangles
In the figure, lines and are parallel. If and , what is the value of ?
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Correct Answer: CChoice C is correct. Vertical angles, which are angles that are opposite each other when two lines intersect, are congruent. The figure shows that lines and intersect. It follows that the angle with measure and the angle with measure are vertical angles, so . It's given that and . Substituting for and for in the equation yields . Subtracting from both sides of this equation yields . Adding to both sides of this equation yields , or . Dividing both sides of this equation by yields . It's given that lines and are parallel, and the figure shows that lines and are intersected by a transversal, line . If two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary. It follows that the same-side interior angles with measures and are supplementary, so . Substituting for in this equation yields . Substituting for in this equation yields , or . Subtracting from both sides of this equation yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of or , not .
Question 127 127 of 269 selected Area & Volume
Parallelogram is similar to parallelogram . The length of each side of parallelogram is times the length of its corresponding side of parallelogram . The area of parallelogram is square centimeters. What is the area, in square centimeters, of parallelogram ?
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Correct Answer: CChoice C is correct. It’s given that parallelogram is similar to parallelogram . When two parallelograms are similar, if the scale factor between their corresponding side lengths is , the scale factor between their areas is . It’s given that the length of each side of parallelogram is times the length of its corresponding side of parallelogram , so the scale factor between their corresponding side lengths is . It follows that the scale factor between their areas is , or . It’s given that the area, in square centimeters, of parallelogram is . It follows that the area, in square centimeters, of parallelogram is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 128 128 of 269 selected Right Triangles & Trigonometry
An isosceles right triangle has a hypotenuse of length inches. What is the perimeter, in inches, of this triangle?
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Correct Answer: CChoice C is correct. Since the triangle is an isosceles right triangle, the two sides that form the right angle must be the same length. Let be the length, in inches, of each of those sides. The Pythagorean theorem states that in a right triangle, , where is the length of the hypotenuse and and are the lengths of the other two sides. Substituting for , for , and for in this equation yields , or . Dividing each side of this equation by yields , or . Taking the square root of each side of this equation yields two solutions: and . The value of must be positive because it represents a side length. Therefore, , or . The perimeter, in inches, of the triangle is , or . Substituting for in this expression gives a perimeter, in inches, of , or .
Choice A is incorrect. This is the length, in inches, of each of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.
Choice B is incorrect. This is the sum of the lengths, in inches, of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 129 129 of 269 selected Circles
A circle has center , and points and lie on the circle. In triangle , the measure of is . What is the measure of , in degrees? (Disregard the degree symbol when entering your answer.)
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Correct Answer: 46The correct answer is . It's given that is the center of a circle and that points and lie on the circle. Therefore, and are radii of the circle. It follows that . If two sides of a triangle are congruent, then the angles opposite them are congruent. It follows that the angles and , which are across from the sides of equal length, are congruent. Let represent the measure of . It follows that the measure of is also . It's given that the measure of is . Because the sum of the measures of the interior angles of a triangle is , the equation , or , can be used to find the measure of . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the measure of , in degrees, is .
Question 130 130 of 269 selected Area & Volume
Rectangles and are similar. The length of each side of is times the length of the corresponding side of . The area of is square units. What is the area, in square units, of ?
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Correct Answer: DChoice D is correct. The area of a rectangle is given by , where is the length of the base of the rectangle and is its height. Let represent the length, in units, of the base of rectangle , and let represent its height, in units. Substituting for and for in the formula yields . Therefore, the area, in square units, of can be represented by the expression . It’s given that the length of each side of is times the length of the corresponding side of . Therefore, the length, in units, of the base of can be represented by the expression , and its height, in units, can be represented by the expression . Substituting for and for in the formula yields , which is equivalent to . Therefore, the area, in square units, of can be represented by the expression . It’s given that the area of is square units. Since represents the area, in square units, of , substituting for in the expression yields , or . Therefore, the area, in square units, of is .
Choice A is incorrect. This is the area of a rectangle where the length of each side of the rectangle is , not , times the length of the corresponding side of .
Choice B is incorrect. This is the area of a rectangle where the length of each side of the rectangle is , not , times the length of the corresponding side of .
Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is , not , times the length of the corresponding side of .
Question 131 131 of 269 selected Right Triangles & Trigonometry
A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. The diagonal of the rectangle is twice the length of the shortest side of the rectangle. The area of the rectangle is square units. What is the length, in units, of the diameter of the circle?
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Correct Answer: 66The correct answer is . It's given that each vertex of the rectangle lies on the circumference of the circle. Therefore, the length of the diameter of the circle is equal to the length of the diagonal of the rectangle. The diagonal of a rectangle forms a right triangle with the shortest and longest sides of the rectangle, where the shortest side and the longest side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. Let represent the length, in units, of the shortest side of the rectangle. Since it's given that the diagonal is twice the length of the shortest side, represents the length, in units, of the diagonal of the rectangle. By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . Substituting for and for in this equation yields , or . Subtracting from both sides of this equation yields . Taking the positive square root of both sides of this equation yields. Therefore, the length, in units, of the rectangle’s longest side is . The area of a rectangle is the product of the length of the shortest side and the length of the longest side. The lengths, in units, of the shortest and longest sides of the rectangle are represented by and , and it’s given that the area of the rectangle is square units. It follows that , or . Dividing both sides of this equation by yields . Taking the positive square root of both sides of this equation yields . Since the length, in units, of the diagonal is represented by , it follows that the length, in units, of the diagonal is , or . Since the length of the diameter of the circle is equal to the length of the diagonal of the rectangle, the length, in units, of the diameter of the circle is .
Question 132 132 of 269 selected Right Triangles & Trigonometry
One leg of a right triangle has a length of millimeters. The hypotenuse of the triangle has a length of millimeters. What is the length of the other leg of the triangle, in millimeters?
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Correct Answer: CChoice C is correct. The Pythagorean theorem states that for a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. It's given that one leg of a right triangle has a length of millimeters. It's also given that the hypotenuse of the triangle has a length of millimeters. Let represent the length of the other leg of the triangle, in millimeters. Therefore, by the Pythagorean theorem, , or . Subtracting from both sides of this equation yields . Taking the positive square root of both sides of this equation yields . Therefore, the length of the other leg of the triangle, in millimeters, is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 133 133 of 269 selected Area & Volume
Triangle ABC and triangle DEF are similar triangles, where and
are corresponding sides. If
and the perimeter of triangle ABC is 20, what is the perimeter of triangle DEF ?
10
40
80
120
Show Answer
Correct Answer: BChoice B is correct. Since triangles ABC and DEF are similar and , the length of each side of triangle DEF is two times the length of its corresponding side in triangle ABC. Therefore, the perimeter of triangle DEF is two times the perimeter of triangle ABC. Since the perimeter of triangle ABC is 20, the perimeter of triangle DEF is 40.
Choice A is incorrect. This is half, not two times, the perimeter of triangle ABC. Choice C is incorrect. This is two times the perimeter of triangle DEF rather than two times the perimeter of triangle ABC. Choice D is incorrect. This is six times, not two times, the perimeter of triangle ABC.
Question 134 134 of 269 selected Circles
In the xy-plane, the graph of is a circle. What is the radius of the circle?
5
6.5
Show Answer
Correct Answer: AChoice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, , where
is the center of the circle and the radius of the circle is r. To do this, divide the original equation,
, by 2 to make the leading coefficients of
and
each equal to 1:
. Then complete the square to put the equation in standard form. To do so, first rewrite
as
. Second, add 2.25 and 0.25 to both sides of the equation:
. Since
,
, and
, it follows that
. Therefore, the radius of the circle is 5.
Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xy-plane.
Question 135 135 of 269 selected Right Triangles & Trigonometry
The perimeter of an equilateral triangle is centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is centimeters. What is the value of ?
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Correct Answer: 284/3, 94.66, 94.67The correct answer is . Since the perimeter of a triangle is the sum of the lengths of its sides, and the given triangle is equilateral, the length of each side is , or , centimeters (cm). Right triangle can be formed, where is the midpoint of one of the triangle’s sides, is one of this side’s endpoints, and is the center of the circle. It follows that is , or , cm. Additionally, triangle has angles measuring , , and , where the measure of angle is and the measure of angle is . It follows that the length of side is half the length of hypotenuse , and the length of side is times the length of side . It’s given that cm. Therefore, cm and cm, which is equivalent to cm. Since cm, it follows that . Multiplying both sides of this equation by yields . Dividing both sides of this equation by yields . Note that 284/3, 94.66, and 94.67 are examples of ways to enter a correct answer.
Question 136 136 of 269 selected Lines, Angles, & Triangles
In triangle , is extended to point . The measure of is , and the measure of is . What is the measure of ?
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Correct Answer: BChoice B is correct. In the figure shown, since is a line segment, the sum of the measures of and is . It's given that the measure of is . Thus, the measure of is , or . The sum of the measures of the interior angles of a triangle is . It's given that the measure of is . Therefore, the measure of is , or .
Choice A is incorrect. This is the measure of the supplement of , not the measure of .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the measure of , not the measure of .
Question 137 137 of 269 selected Right Triangles & Trigonometry
An isosceles right triangle has a perimeter of inches. What is the length, in inches, of one leg of this triangle?
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Correct Answer: BChoice B is correct. It's given that the right triangle is isosceles. In an isosceles right triangle, the two legs have equal lengths, and the length of the hypotenuse is times the length of one of the legs. Let represent the length, in inches, of each leg of the isosceles right triangle. It follows that the length of the hypotenuse is inches. The perimeter of a figure is the sum of the lengths of the sides of the figure. Therefore, the perimeter of the isosceles right triangle is inches. It's given that the perimeter of the triangle is inches. It follows that . Factoring the left-hand side of this equation yields , or . Dividing both sides of this equation by yields . Rationalizing the denominator of the right-hand side of this equation by multiplying the right-hand side of the equation by yields . Applying the distributive property to the numerator and to the denominator of the right-hand side of this equation yields . This is equivalent to , or . Therefore, the length, in inches, of one leg of the isosceles right triangle is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the length, in inches, of the hypotenuse.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 138 138 of 269 selected Lines, Angles, & Triangles
Quadrilaterals and are similar, where , , and correspond to , , and , respectively. The measure of is , , and . What is the measure of ?
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Correct Answer: DChoice D is correct. Corresponding angles in similar figures have equal measure. It's given that quadrilaterals and are similar and that , , and correspond to , , and . It follows that corresponds to . It's also given that the measure of is . Therefore, the measure of is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect. This is the supplement of the measure of , not the measure of .
Question 139 139 of 269 selected Right Triangles & Trigonometry
The side lengths of right triangle are given. Triangle is similar to triangle , where corresponds to and corresponds to . What is the value of ?
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Correct Answer: DChoice D is correct. The hypotenuse of triangle is the longest side and is across from the right angle. The longest side length given is , which is the length of side . Therefore, the hypotenuse of triangle is side , so the right angle is angle . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side, which is the side across from the angle, to the length of the adjacent side, which is the side closest to the angle that is not the hypotenuse. It follows that the opposite side of angle is side and the adjacent side of angle is side . Therefore, . Substituting for and for in this equation yields . This is equivalent to . It’s given that triangle is similar to triangle , where corresponds to and corresponds to . It follows that corresponds to . Therefore, the hypotenuse of triangle is side , which means . Since the lengths of corresponding sides of similar triangles are proportional, . Therefore, is equivalent to , or . Thus, .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
Question 140 140 of 269 selected Right Triangles & Trigonometry
In the right triangle shown, what is the value of ?
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Correct Answer: CChoice C is correct. The sine of an acute angle in a right triangle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. In the right triangle shown, it's given that the length of the side opposite the angle with measure is units and the length of the hypotenuse is units. Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 141 141 of 269 selected Lines, Angles, & Triangles
The area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?
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The correct answer is either ,
, or
. The area of triangle ABC can be expressed as
or
. It’s given that the area of triangle ABC is at least 48 but no more than 60. It follows that
. Dividing by 6 to isolate y in this compound inequality yields
. Since y is an integer,
. In the given figure, the two right triangles shown are similar because they have two pairs of congruent angles: their respective right angles and angle A. Therefore, the following proportion is true:
. Substituting 8 for y in the proportion results in
. Cross multiplying and solving for x yields
. Substituting 9 for y in the proportion results in
. Cross multiplying and solving for x yields
. Substituting 10 for y in the proportion results in
. Cross multiplying and solving for x yields
. Note that 10/3, 15/4, 25/6, 3.333, 3.75, 4.166, and 4.167 are examples of ways to enter a correct answer.
Question 142 142 of 269 selected Area & Volume
A rectangular poster has an area of square inches. A copy of the poster is made in which the length and width of the original poster are each increased by . What is the area of the copy, in square inches?
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Correct Answer: 2592/5, 518.4The correct answer is . It's given that the area of the original poster is square inches. Let represent the length, in inches, of the original poster, and let represent the width, in inches, of the original poster. Since the area of a rectangle is equal to its length times its width, it follows that . It's also given that a copy of the poster is made in which the length and width of the original poster are each increased by . It follows that the length of the copy is the length of the original poster plus of the length of the original poster, which is equivalent to inches. This length can be rewritten as inches, or inches. Similarly, the width of the copy is the width of the original poster plus of the width of the original poster, which is equivalent to inches. This width can be rewritten as inches, or inches. Since the area of a rectangle is equal to its length times its width, it follows that the area, in square inches, of the copy is equal to , which can be rewritten as . Since , the area, in square inches, of the copy can be found by substituting for in the expression , which yields , or . Therefore, the area of the copy, in square inches, is .
Question 143 143 of 269 selected Circles
What is the center of the circle in the xy-plane defined by the equation ?
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Correct Answer: CChoice C is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is and the radius of the circle is . It's given that the circle in the xy-plane is defined by the equation . This equation can be written as . For this equation, it follows that and . Therefore, the center of the circle in the xy-plane defined by the given equation is .
Choice A is incorrect. This is the center of the circle in the xy-plane that is defined by the equation , not .
Choice B is incorrect. This is the center of the circle in the xy-plane that is defined by the equation , not .
Choice D is incorrect. This is the center of the circle in the xy-plane that is defined by the equation , not .
Question 144 144 of 269 selected Area & Volume
A right rectangular prism has a length of , a width of , and a height of . What is the surface area, , of the right rectangular prism?
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Correct Answer: 2216The correct answer is . The surface area of a prism is the sum of the areas of all its faces. A right rectangular prism consists of six rectangular faces, where opposite faces are congruent. It's given that this prism has a length of , a width of , and a height of . Thus, for this prism, there are two faces with area , two faces with area , and two faces with area . Therefore, the surface area, , of the right rectangular prism is , or .
Question 145 145 of 269 selected Circles
A circle in the xy-plane has its center at and has a radius of . Which equation represents this circle?
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Correct Answer: BChoice B is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is and the radius of the circle is . It’s given that this circle has a center at and a radius of . Substituting for , for , and for in yields , or . Therefore, the equation that represents this circle is .
Choice A is incorrect. This equation represents a circle with radius , not .
Choice C is incorrect. This equation represents a circle with radius , not .
Choice D is incorrect. This equation represents a circle with radius , not .
Question 146 146 of 269 selected Area & Volume
What is the area, in square centimeters, of a rectangle with a length of and a width of ?
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Correct Answer: 986The correct answer is . The area, , of a rectangle is given by , where is the length of the rectangle and is its width. It’s given that the length of the rectangle is centimeters (cm) and the width is cm. Substituting for and for in the equation yields , or . Therefore, the area, in square centimeters, of this rectangle is .
Question 147 147 of 269 selected Circles
An angle has a measure of radians. What is the measure of the angle in degrees?
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Correct Answer: 81The correct answer is . The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by . Multiplying the given angle measure, radians, by yields , which is equivalent to degrees.
Question 148 148 of 269 selected Area & Volume
The lengths of two sides of a triangle are centimeters and centimeters. If the perimeter of the triangle is centimeters, what is the length, in centimeters, of the third side of this triangle?
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Correct Answer: BChoice B is correct. The perimeter of a triangle is the sum of the lengths of all three of its sides. It's given that the lengths of two sides of a triangle are centimeters and centimeters. Let represent the length, in centimeters, of the third side of this triangle. The sum of the lengths, in centimeters, of all three sides of the triangle can be represented by the expression . Since it’s given that the perimeter of the triangle is centimeters, it follows that , or . Subtracting from both sides of this equation yields . Therefore, the length, in centimeters, of the third side of this triangle is .
Choice A is incorrect. If the length of the third side of this triangle were centimeters, the perimeter, in centimeters, of the triangle would be , or , not .
Choice C is incorrect. If the length of the third side of this triangle were centimeters, the perimeter, in centimeters, of the triangle would be , or , not .
Choice D is incorrect. If the length of the third side of this triangle were centimeters, the perimeter, in centimeters, of the triangle would be , or , not .
Question 149 149 of 269 selected Circles
In the xy-plane, the graph of the given equation is a circle. What are the coordinates of the center of the circle?
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Correct Answer: DChoice D is correct. It’s given that in the xy-plane, the graph of is a circle. The equation of a circle in the xy-plane can be written as , where the coordinates of the center of the circle are and the radius of the circle is . By completing the square, the equation can be rewritten as , or . This equation is equivalent to , or . Therefore, is and is , and the coordinates of the center of the circle are .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 150 150 of 269 selected Area & Volume
A right circular cone has a volume of cubic feet and a height of 9 feet. What is the radius, in feet, of the base of the cone?
3
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Correct Answer: AChoice A is correct. The equation for the volume of a right circular cone is . It’s given that the volume of the right circular cone is
cubic feet and the height is 9 feet. Substituting these values for V and h, respectively, gives
. Dividing both sides of the equation by
gives
. Dividing both sides of the equation by 9 gives
. Taking the square root of both sides results in two possible values for the radius,
or
. Since the radius can’t have a negative value, that leaves
as the only possibility. Applying the quotient property of square roots,
, results in
, or
.
Choices B and C are incorrect and may result from incorrectly evaluating . Choice D is incorrect and may result from solving
instead of
.
Question 151 151 of 269 selected Lines, Angles, & Triangles
In the triangle above, . What is the value of b ?
52
59
76
104
Show Answer
Correct Answer: AChoice A is correct. The sum of the measures of the three interior angles of a triangle is 180°. Therefore, . Since it’s given that
, it follows that
, or
. Dividing both sides of this equation by 2 yields
.
Choice B is incorrect and may result from a calculation error. Choice C is incorrect. This is the value of . Choice D is incorrect. This is the value of
.
Question 152 152 of 269 selected Circles
An angle has a measure of radians. What is the measure of the angle, in degrees?
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Correct Answer: 192The correct answer is . The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by . Multiplying the given angle measure, , by yields , which simplifies to degrees.
Question 153 153 of 269 selected Circles
Circle A shown is defined by the equation . Circle B (not shown) has the same radius but is translated units to the right. If the equation of circle B is , where , , and are constants, what is the value of ?
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Correct Answer: 28The correct answer is . The equation of a circle in the xy-plane can be written as , where the center of the circle is and the radius of the circle is . It’s given that circle A is defined by the equation , which can be written as . It follows that and the radius of circle A is . It’s also given that circle B has the same radius as circle A. If the equation of circle B is , then . Substituting for in this equation yields , or . It follows that the value of is , or .
Question 154 154 of 269 selected Right Triangles & Trigonometry
A right triangle has legs with lengths of centimeters and centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form , where is an integer, what is the value of ?
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Correct Answer: 113The correct answer is . It's given that the legs of a right triangle have lengths centimeters and centimeters. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. It follows that if represents the length, in centimeters, of the hypotenuse of the right triangle, . This equation is equivalent to . Taking the square root of each side of this equation yields . This equation can be rewritten as , or . This equation is equivalent to . It's given that the length of the triangle's hypotenuse, in centimeters, can be written in the form . It follows that the value of is .
Question 155 155 of 269 selected Right Triangles & Trigonometry
The side lengths of right triangle are given. Triangle is similar to triangle , where corresponds to and corresponds to . What is the value of ?
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Correct Answer: BChoice B is correct. It's given that right triangle is similar to triangle , where corresponds to and corresponds to . It's given that the side lengths of the right triangle are , , and . Corresponding angles in similar triangles are equal. It follows that the measure of angle is equal to the measure of angle . The hypotenuse of a right triangle is the longest side. It follows that the hypotenuse of triangle is side . The hypotenuse of a right triangle is the side opposite the right angle. Therefore, angle is a right angle. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle is side . The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle is side . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, . Substituting for and for in this equation yields , or . The tangents of two acute angles with equal measures are equal. Since the measure of angle is equal to the measure of angle , it follows that . Substituting for in this equation yields . Therefore, the value of is .
Choice A is incorrect. This is the value of .
Choice C is incorrect. This is the value of .
Choice D is incorrect. This is the value of .
Question 156 156 of 269 selected Lines, Angles, & Triangles
In the figure, two lines intersect at a point. Angle and angle are vertical angles. The measure of angle is . What is the measure of angle ?
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Correct Answer: AChoice A is correct. It’s given that angle and angle are vertical angles, and the measure of angle is . Vertical angles have equal measures. Therefore, the measure of angle is .
Choice B is incorrect. This is the measure of an angle that is supplementary, not congruent, to angle .
Choice C is incorrect. This is the sum of the measures of angle and angle .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 157 157 of 269 selected Lines, Angles, & Triangles
In the figure, three lines intersect at point P. If and
, what is the value of z ?
140
80
40
20
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Correct Answer: CChoice C is correct. The angle that is shown as lying between the y° angle and the z° angle is a vertical angle with the x° angle. Since vertical angles are congruent and , the angle between the y° angle and the z° angle measures 65°. Since the 65° angle, the y° angle, and the z° angle are adjacent and form a straight angle, it follows that the sum of the measures of these three angles is 180°, which is represented by the equation
. It’s given that y = 75. Substituting 75 for y yields
, which can be rewritten as
. Subtracting 140° from both sides of this equation yields
. Therefore,
.
Choice A is incorrect and may result from finding the value of rather than z. Choices B and D are incorrect and may result from conceptual or computational errors.
Question 158 158 of 269 selected Area & Volume
A circle has a radius of inches. The area of the circle is square inches, where is a constant. What is the value of ?
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Correct Answer: 4.41, 441/100The correct answer is . The area, , of a circle is given by the formula , where is the radius of the circle. It's given that the area of the circle is square inches, where is a constant, and the radius of the circle is inches. Substituting for and for in the formula yields . Dividing both sides of this equation by yields . Therefore, the value of is .
Question 159 159 of 269 selected Area & Volume
A rectangle has a length of and a width of . What is the perimeter of the rectangle?
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Correct Answer: CChoice C is correct. The perimeter of a quadrilateral is the sum of the lengths of its four sides. It's given that the rectangle has a length of and a width of . It follows that the rectangle has two sides with length and two sides with length . Therefore, the perimeter of the rectangle is , or .
Choice A is incorrect. This is the sum of the lengths of the two sides with length , not the sum of the lengths of all four sides of the rectangle.
Choice B is incorrect. This is the sum of the lengths of the two sides with length , not the sum of the lengths of all four sides of the rectangle.
Choice D is incorrect. This is the perimeter of a rectangle that has four sides with length , not two sides with length and two sides with length .
Question 160 160 of 269 selected Right Triangles & Trigonometry
In triangle RST above, point W (not shown) lies on . What is the value of
?
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The correct answer is 0. Note that no matter where point W is on , the sum of the measures of
and
is equal to the measure of
, which is
. Thus,
and
are complementary angles. Since the cosine of an angle is equal to the sine of its complementary angle,
. Therefore,
.
Question 161 161 of 269 selected Lines, Angles, & Triangles
In the figure shown, line is parallel to line . What is the value of ?
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Correct Answer: DChoice D is correct. The sum of consecutive interior angles between two parallel lines and on the same side of the transversal is degrees. Since it's given that line is parallel to line , it follows that . Subtracting from both sides of this equation yields . Therefore, the value of is .
Choice A is incorrect. This is half of the given angle measure.
Choice B is incorrect. This is the value of the given angle measure.
Choice C is incorrect. This is twice the value of the given angle measure.
Question 162 162 of 269 selected Right Triangles & Trigonometry
A right triangle has legs with lengths of centimeters and centimeters. What is the length of this triangle's hypotenuse, in centimeters?
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Correct Answer: BChoice B is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. It's given that the right triangle has legs with lengths of centimeters and centimeters. Let represent the length of this triangle's hypotenuse, in centimeters. Therefore, by the Pythagorean theorem, , or . Taking the positive square root of both sides of this equation yields , or . Therefore, the length of this triangle's hypotenuse, in centimeters, is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the square of the length of the triangle’s hypotenuse.
Question 163 163 of 269 selected Lines, Angles, & Triangles
In the figure above, lines and k are parallel. What is the value of a ?
26
64
116
154
Show Answer
Correct Answer: CChoice C is correct. Since lines and k are parallel, corresponding angles formed by the intersection of line j with lines
and k are congruent. Therefore, the angle with measure a° must be the supplement of the angle with measure 64°. The sum of two supplementary angles is 180°, so a = 180 – 64 = 116.
Choice A is incorrect and likely results from thinking the angle with measure a° is the complement of the angle with measure 64°. Choice B is incorrect and likely results from thinking the angle with measure a° is congruent to the angle with measure 64°. Choice D is incorrect and likely results from a conceptual or computational error.
Question 164 164 of 269 selected Lines, Angles, & Triangles
In triangle , the measure of angle is and the measure of angle is . What is the measure of angle ?
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Correct Answer: DChoice D is correct. The sum of the angle measures of a triangle is . Adding the measures of angles and gives . Therefore, the measure of angle is .
Choice A is incorrect and may result from subtracting the sum of the measures of angles and from , instead of from .
Choice B is incorrect and may result from subtracting the measure of angle from the measure of angle .
Choice C is incorrect and may result from adding the measures of angles and but not subtracting the result from .
Question 165 165 of 269 selected Circles
The circle above with center O has a circumference of 36. What is the length of minor arc ?
9
12
18
36
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Correct Answer: AChoice A is correct. A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and is a central angle of the circle. From the figure, the two diameters that meet to form
are perpendicular, so the measure of
is
. Therefore, the length of minor arc
is
of the circumference of the circle. Since the circumference of the circle is 36, the length of minor arc
is
.
Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc is
of the circumference. However, the lengths in choices B and C are, respectively,
and
the circumference of the circle, and the length in choice D is the length of the entire circumference. None of these lengths is
the circumference.
Question 166 166 of 269 selected Circles
The measure of angle is radians. The measure of angle is radians greater than the measure of angle . What is the measure of angle , in degrees?
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Correct Answer: CChoice C is correct. It’s given that the measure of angle is radians, and the measure of angle is radians greater than the measure of angle . Therefore, the measure of angle is equal to radians. Multiplying by to get a common denominator with yields . Therefore, is equivalent to , or . Therefore, the measure of angle is radians. The measure of angle , in degrees, can be found by multiplying its measure, in radians, by . This yields , which is equivalent to degrees. Therefore, the measure of angle is degrees.
Choice A is incorrect. This is the number of degrees that the measure of angle is greater than the measure of angle .
Choice B is incorrect. This is the measure of angle , in degrees.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 167 167 of 269 selected Right Triangles & Trigonometry
Triangle is similar to triangle , where corresponds to and corresponds to . Angles and are right angles. If and , what is the length of ?
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Correct Answer: DChoice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle is similar to triangle , where corresponds to , so the measure of angle is equal to the measure of angle . Therefore, if , then . It's given that angles and are right angles, so triangles and are right triangles. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle is side . The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle is side . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, . If , the length of side can be found by substituting for and for in the equation , which yields . Multiplying both sides of this equation by yields . Since the length of side is times the length of side , it follows that triangle is a special right triangle with angle measures , , and . Therefore, the length of the hypotenuse, , is times the length of side , or . Substituting for in this equation yields , or . Thus, if and , the length of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the length of , not .
Question 168 168 of 269 selected Lines, Angles, & Triangles
Triangle ABC and triangle DEF are shown. The relationship between the side lengths of the two triangles is such that . If the measure of angle BAC is 20°, what is the measure, in degrees, of angle EDF ? (Disregard the degree symbol when gridding your answer.)
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The correct answer is 20. By the equality given, the three pairs of corresponding sides of the two triangles are in the same proportion. By the side-side-side (SSS) similarity theorem, triangle ABC is similar to triangle DEF. In similar triangles, the measures of corresponding angles are congruent. Since angle BAC corresponds to angle EDF, these two angles are congruent and their measures are equal. It’s given that the measure of angle BAC is 20°, so the measure of angle EDF is also 20°.
Question 169 169 of 269 selected Circles
The measure of angle is . What is the measure, in radians, of angle ?
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Correct Answer: BChoice B is correct. The measure of an angle, in radians, can be found by multiplying its measure, in degrees, by . It's given that the measure of angle is . It follows that the measure, in radians, of angle is , or .
Choice A is incorrect. This is the measure, in radians, of an angle whose measure is , not .
Choice C is incorrect. This is the measure, in radians, of an angle whose measure is , not .
Choice D is incorrect. This is the measure, in radians, of an angle whose measure is , not .
Question 170 170 of 269 selected Circles
The equation represents circle A. Circle B is obtained by shifting circle A down units in the xy-plane. Which of the following equations represents circle B?
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Correct Answer: DChoice D is correct. The graph in the xy-plane of an equation of the form is a circle with center and a radius of length . It's given that circle A is represented by , which can be rewritten as . Therefore, circle A has center and a radius of length . Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point on circle A, the point lies on circle B. Moreover, if is the center of circle A, then is the center of circle B. Therefore, circle B has a radius of and the center of circle B is , or . Thus, circle B can be represented by the equation , or .
Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right units.
Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up units.
Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left units.
Question 171 171 of 269 selected Area & Volume
Rectangles and shown are similar, where , , , and correspond to , , , and , respectively. What is the length, in inches , of ?
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Correct Answer: BChoice B is correct. It’s given that rectangles and are similar, where , , , and correspond to , , , and , respectively. It follows that corresponds to and corresponds to . If two rectangles are similar, then the lengths of their corresponding sides are proportional. It’s given in the figure that the length of is inches, the length of is inches, and the length of is inches. If is the length, in inches, of , then is equivalent to . Therefore, the value of can be found using the equation . Multiplying each side of this equation by yields , or . Therefore, the length, in inches, of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the length, in inches, of , not .
Question 172 172 of 269 selected Lines, Angles, & Triangles
In triangle , angle is a right angle, point lies on , and point lies on such that is parallel to . If the measure of angle is , what is the measure, in degrees, of angle ?
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Correct Answer: 153The correct answer is . Since it's given that is parallel to and angle is a right angle, angle is also a right angle. Angle is complementary to angle , which means its measure, in degrees, is , or . Since angle is supplementary to angle , its measure, in degrees, is , or .
Question 173 173 of 269 selected Circles
What is the value of ?
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Correct Answer: AChoice A is correct. A trigonometric ratio can be found using the unit circle, that is, a circle with radius unit. If a central angle of a unit circle in the xy-plane centered at the origin has its starting side on the positive x-axis and its terminal side intersects the circle at a point , then the value of the tangent of the central angle is equal to the y-coordinate divided by the x-coordinate. There are radians in a circle. Dividing by yields , which is equivalent to . It follows that on the unit circle centered at the origin in the xy-plane, the angle is the result of revolutions from its starting side on the positive x-axis followed by a rotation through radians. Therefore, the angles and are coterminal angles and is equal to . Since is greater than and less than , it follows that the terminal side of the angle is in quadrant II and forms an angle of , or , with the negative x-axis. Therefore, the terminal side of the angle intersects the unit circle at the point . It follows that the value of is , which is equivalent to . Therefore, the value of is .
Choice B is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
Question 174 174 of 269 selected Lines, Angles, & Triangles
Triangles and are graphed in the xy-plane. Triangle has vertices , , and at , , and , respectively. Triangle has vertices , , and at , , and , respectively, where is a positive constant. If the measure of is , what is the measure of ?
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Correct Answer: CChoice C is correct. Since and , side is parallel to the y-axis and has a length of . Since and , side is parallel to the x-axis and has a length of . Therefore, triangle is a right isosceles triangle, where has measure and and each have measure . It follows that if the measure of is , then . Since and , side is parallel to the y-axis and has a length of . Since and , side is parallel to the x-axis and has a length of . Therefore, triangle is a right isosceles triangle, where has measure and and each have measure . Of the given choices, only is equal to , so the measure of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 175 175 of 269 selected Area & Volume
A triangular prism has a height of and a volume of . What is the area, , of the base of the prism? (The volume of a triangular prism is equal to , where is the area of the base and is the height of the prism.)
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Correct Answer: 27The correct answer is . It's given that a triangular prism has a volume of and the volume of a triangular prism is equal to , where is the area of the base and is the height of the prism. Therefore, . It's also given that the triangular prism has a height of . Therefore, . Substituting for in the equation yields . Dividing both sides of this equation by yields . Therefore, the area, , of the base of the prism is .
Question 176 176 of 269 selected Lines, Angles, & Triangles
In the given figure, extends to point D. If the measure of
is equal to the measure of
, what is the value of x ?
110
70
55
40
Show Answer
Correct Answer: DChoice D is correct. Since and
form a linear pair of angles, their measures sum to 180°. It’s given that the measure of
is 110°. Therefore,
. Subtracting 110° from both sides of this equation gives the measure of
as 70°. It’s also given that the measure of
is equal to the measure of
. Thus, the measure of
is also 70°. The measures of the interior angles of a triangle sum to 180°. Thus,
. Combining like terms on the left-hand side of this equation yields
. Subtracting 140° from both sides of this equation yields
, or
.
Choice A is incorrect. This is the value of the measure of . Choice B is incorrect. This is the value of the measure of each of the other two interior angles,
and
. Choice C is incorrect and may result from an error made when identifying the relationship between the exterior angle of a triangle and the interior angles of the triangle.
Question 177 177 of 269 selected Circles
A circle has center , and points and lie on the circle. Line segments and are tangent to the circle at points and , respectively. If the radius of the circle is millimeters and the perimeter of quadrilateral is millimeters, what is the distance, in millimeters, between points and ?
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Correct Answer: DChoice D is correct. It's given that the radius of the circle is millimeters. Since points and both lie on the circle, segments and are both radii. Therefore, segments and each have length millimeters. Two segments that are tangent to a circle and have a common exterior endpoint have equal length. Therefore, segment and segment have equal length. Let represent the length of segment . Then also represents the length of segment . It's given that the perimeter of quadrilateral is millimeters. Since the perimeter of a quadrilateral is equal to the sum of the lengths of the sides of the quadrilateral, , or . Subtracting from both sides of this equation yields , and dividing both sides of this equation by yields . Therefore, the length of segment is millimeters. A line segment that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. Therefore, segment is perpendicular to segment . Since perpendicular segments form right angles, angle is a right angle. Therefore, triangle is a right triangle with legs of length millimeters and millimeters, and hypotenuse . By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . Substituting for and for in this equation yields , or . Taking the square root of both sides of this equation yields . Since represents a length, which must be positive, the value of is . Therefore, the length of segment is millimeters, so the distance between points and is millimeters.
Choice A is incorrect. This is the distance between points and and between points and , not the distance between points and .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the distance between points and and between points and , not the distance between points and .
Question 178 178 of 269 selected Lines, Angles, & Triangles
Triangles and are congruent, where corresponds to , and and are right angles. The measure of angle is . What is the measure, in degrees, of angle ?
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Correct Answer: 21The correct answer is . It's given that triangles and are congruent with angle corresponding to angle . Corresponding angles of congruent triangles are congruent and, therefore, have equal measure. It's given that the measure of angle is . It follows that the measure of angle is also . It's given that angle is a right angle. Therefore, the measure of angle is . Let represent the measure, in degrees, of angle . Since the measures of the angles in a triangle sum to , it follows that , or . Subtracting from both sides of this equation yields . Therefore, the measure, in degrees, of angle is .
Question 179 179 of 269 selected Area & Volume
Each base of a right rectangular prism has a length of inches and a width of inches. The prism has a volume of cubic inches. What is the height, in inches, of the prism?
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Correct Answer: AChoice A is correct. The volume, , of a rectangular prism is given by the formula , where is the length of the base, is the width of the base, and is the height of the prism. It’s given that each base of a right rectangular prism has a length of inches and a width of inches, and that the prism has a volume of cubic inches. Substituting for , for , and for in the formula gives , or . Dividing each side of this equation by yields . Therefore, the height, in inches, of the prism is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the area, in square inches, of the base of the prism, not the height, in inches, of the prism.
Question 180 180 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line . What is the value of ?
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Correct Answer: 47The correct answer is . Based on the figure, the angle with measure and the angle with measure together form a straight line. Therefore, these two angles are supplementary, so the sum of their measures is . It follows that . Subtracting from both sides of this equation yields .
Question 181 181 of 269 selected Area & Volume
The triangle shown has a perimeter of units. If units and units, what is the value of , in units?
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Correct Answer: AChoice A is correct. The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths , , and . It's given that the triangle has a perimeter of units. Therefore, . If units and units, the value of , in units, can be found by substituting for and for in the equation , which yields , or . Subtracting from both sides of this equation yields . Therefore, if units and units, the value of , in units, is .
Choice B is incorrect. This is the value of , in units, not the value of , in units.
Choice C is incorrect. This is the value of , in units, not the value of , in units.
Choice D is incorrect. This is the value of , in units, not the value of , in units.
Question 182 182 of 269 selected Lines, Angles, & Triangles
In triangle , angle is a right angle, point lies on , point lies on , and is parallel to . If the length of is units, the length of is units, and the area of triangle is square units, what is the length of , in units?
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Correct Answer: 14.66, 14.67, 44/3The correct answer is . It's given that in triangle , angle is a right angle. The area of a right triangle can be found using the formula , where represents the area of the right triangle, represents the length of one leg of the triangle, and represents the length of the other leg of the triangle. In triangle , the two legs are and . Therefore, if the length of is and the area of triangle is , then , or . Dividing both sides of this equation by yields . Therefore, the length of is . It's also given that point lies on , point lies on , and is parallel to . It follows that angle is a right angle. Since triangles and share angle and have right angles and , respectively, triangles and are similar triangles. Therefore, the ratio of the length of to the length of is equal to the ratio of the length of to the length of . If the length of is and the length of is , it follows that the ratio of the length of to the length of is , or , so the ratio of the length of to the length of is . Therefore, . Multiplying both sides of this equation by yields . Dividing both sides of this equation by yields . Since the length of , , is the sum of the length of , , and the length of , it follows that the length of is , or . Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.
Question 183 183 of 269 selected Circles
A circle has center , and points and lie on the circle. The measure of arc is and the length of arc is inches. What is the circumference, in inches, of the circle?
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Correct Answer: DChoice D is correct. It’s given that the measure of arc is and the length of arc is . The arc measure of the full circle is . If represents the circumference, in inches, of the circle, it follows that . This equation is equivalent to , or . Multiplying both sides of this equation by yields , or . Therefore, the circumference of the circle is .
Choice A is incorrect. This is the length of arc .
Choice B is incorrect and may result from multiplying the length of arc by .
Choice C is incorrect and may result from squaring the length of arc .
Question 184 184 of 269 selected Circles
In the xy-plane, a circle with radius 5 has center . Which of the following is an equation of the circle?
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Correct Answer: BChoice B is correct. An equation of a circle is , where the center of the circle is
and the radius is r. It’s given that the center of this circle is
and the radius is 5. Substituting these values into the equation gives
, or
.
Choice A is incorrect. This is an equation of a circle that has center . Choice C is incorrect. This is an equation of a circle that has center
and radius
. Choice D is incorrect. This is an equation of a circle that has radius
.
Question 185 185 of 269 selected Lines, Angles, & Triangles
In the figure shown, line intersects parallel lines and . What is the value of ?
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Correct Answer: 70The correct answer is . Based on the figure, the angle with measure and the angle vertical to the angle with measure are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure also has measure . It’s given that lines and are parallel. Therefore, same side interior angles between lines and are supplementary. It follows that . Subtracting from both sides of this equation yields .
Question 186 186 of 269 selected Lines, Angles, & Triangles
Triangles and are congruent, where corresponds to , and and are right angles. The measure of angle is . What is the measure of angle ?
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Correct Answer: BChoice B is correct. It’s given that triangle is congruent to triangle . Corresponding angles of congruent triangles are congruent and, therefore, have equal measure. It’s given that angle corresponds to angle , and that the measure of angle is . It's also given that the measures of angles and are . Since these angles have equal measure, they are corresponding angles. It follows that angle corresponds to angle . Let represent the measure of angle . Since the sum of the measures of the interior angles of a triangle is , it follows that , or . Subtracting from both sides of this equation yields . Therefore, the measure of angle is . Since angle corresponds to angle , it follows that the measure of angle is also .
Choice A is incorrect. This is the measure of angle , not the measure of angle .
Choice C is incorrect. This is the measure of angle , not the measure of angle .
Choice D is incorrect. This is the sum of the measures of angles and , not the measure of angle .
Question 187 187 of 269 selected Area & Volume
A right circular cone has a volume of cubic centimeters and the area of its base is square centimeters. What is the slant height, in centimeters, of this cone?
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Correct Answer: DChoice D is correct. The volume, , of a right circular cone is given by the formula , where is the area of the circular base of the cone and is the height. It’s given that this right circular cone has a volume of cubic centimeters and the area of its base is square centimeters. Substituting for and for in the formula yields . Dividing each side of this equation by yields . Multiplying each side of this equation by yields . Let represent the slant height, in centimeters, of this cone. A right triangle is formed by the radius, , height, , and slant height, , of this cone, where and are the legs of the triangle and is the hypotenuse. Using the Pythagorean theorem, the equation represents this relationship. Because is the area of the base and the area of the base is , it follows that . Dividing both sides of this equation by yields . Substituting for and for in the equation yields , which is equivalent to , or . Taking the positive square root of both sides of this equation yields . Therefore, the slant height of the cone is centimeters.
Choice A is incorrect. This is one-third of the height, in centimeters, not the slant height, in centimeters, of this cone.
Choice B is incorrect. This is the height, in centimeters, not the slant height, in centimeters, of this cone.
Choice C is incorrect. This is the radius, in centimeters, of the base, not the slant height, in centimeters, of this cone.
Question 188 188 of 269 selected Area & Volume
A triangle has a base length of centimeters and a corresponding height of centimeters. What is the area, in square centimeters, of the triangle?
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Correct Answer: BChoice B is correct. The area, , of a triangle is given by , where is the length of a base of the triangle and is the corresponding height of the triangle. It's given that a triangle has a base length of centimeters and a corresponding height of centimeters. Substituting for and for in the formula yields , or . Therefore, the area, in square centimeters, of the triangle is .
Choice A is incorrect. This is the product of the given base and height of the triangle, not its area.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the sum of the given base and height of the triangle, not its area.
Question 189 189 of 269 selected Circles
Which of the following equations represents a circle in the xy-plane that intersects the y-axis at exactly one point?
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Correct Answer: CChoice C is correct. The graph of the equation in the xy-plane is a circle with center and a radius of length . The radius of a circle is the distance from the center of the circle to any point on the circle. If a circle in the xy-plane intersects the y-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on the y-axis must be equal to the length of the circle's radius. It follows that the x-coordinate of the circle's center must be equivalent to the length of the circle's radius. In other words, if the graph of is a circle that intersects the y-axis at exactly one point, then must be true. The equation in choice C is , or . This equation is in the form , where , , and , and represents a circle in the xy-plane with center and radius of length . Substituting for and for in the equation yields , or , which is true. Therefore, the equation in choice C represents a circle in the xy-plane that intersects the y-axis at exactly one point.
Choice A is incorrect. This is the equation of a circle that does not intersect the y-axis at any point.
Choice B is incorrect. This is an equation of a circle that intersects the x-axis, not the y-axis, at exactly one point.
Choice D is incorrect. This is the equation of a circle with the center located on the y-axis and thus intersects the y-axis at exactly two points, not exactly one point.
Question 190 190 of 269 selected Right Triangles & Trigonometry
A right triangle has legs with lengths of centimeters and centimeters. What is the length of this triangle's hypotenuse, in centimeters?
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Correct Answer: BChoice B is correct. The Pythagorean theorem states that for a right triangle, , where represents the length of the hypotenuse and and represent the lengths of the legs. It’s given that a right triangle has legs with lengths of centimeters and centimeters. Substituting for and for in the formula yields , which is equivalent to , or . Taking the square root of each side of this equation yields . Since represents a length, must be positive. Therefore, the length of the triangle’s hypotenuse, in centimeters, is .
Choice A is incorrect. This is the result of solving the equation , not .
Choice C is incorrect. This is the result of solving the equation , not .
Choice D is incorrect. This is the result of solving the equation , not .
Question 191 191 of 269 selected Right Triangles & Trigonometry
In above, what is the length of
?
4
6
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Correct Answer: BChoice B is correct. Triangles ADB and CDB are both triangles and share
. Therefore, triangles ADB and CDB are congruent by the angle-side-angle postulate. Using the properties of
triangles, the length of
is half the length of hypotenuse
. Since the triangles are congruent,
. So the length of
is
.
Alternate approach: Since angle CBD has a measure of , angle ABC must have a measure of
. It follows that triangle ABC is equilateral, so side AC also has length 12. It also follows that the altitude BD is also a median, and therefore the length of AD is half of the length of AC, which is 6.
Choice A is incorrect. If the length of were 4, then the length of
would be 8. However, this is incorrect because
is congruent to
, which has a length of 12. Choices C and D are also incorrect. Following the same procedures as used to test choice A gives
a length of
for choice C and
for choice D. However, these results cannot be true because
is congruent to
, which has a length of 12.
Question 192 192 of 269 selected Area & Volume
What is the area, in square centimeters, of a rectangle with a length of centimeters and a width of centimeters?
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Correct Answer: DChoice D is correct. The area , in square centimeters, of a rectangle can be found using the formula , where is the length, in centimeters, of the rectangle and is its width, in centimeters. It's given that the rectangle has a length of centimeters and a width of centimeters. Substituting for and for in the formula yields , or . Therefore, the area, in square centimeters, of this rectangle is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the perimeter, in centimeters, not the area, in square centimeters, of the rectangle.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 193 193 of 269 selected Area & Volume
The length of each side of a square is centimeters (cm). Which expression gives the area, in , of the square?
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Correct Answer: DChoice D is correct. The area of a square is given by , where is the length of each side of the square. It's given that the length of each side of a square is . It follows that the area, in , of the square is , or . Therefore, the expression that gives the area, in , of the square is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect. This expression gives the perimeter, in , of the square.
Question 194 194 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line , and line intersects both lines. Which of the following statements is true?
The value of is less than .
The value of is greater than .
The value of is equal to .
The value of cannot be determined.
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Correct Answer: CChoice C is correct. Vertical angles, or angles that are opposite each other when two lines intersect, are congruent. It’s given that line intersects line . Based on the figure, the angle with measure and the angle with measure are vertical angles. Therefore, the value of is equal to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 195 195 of 269 selected Circles
In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates . If
is a diameter of the circle, what are the coordinates of point Q ?
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Correct Answer: AChoice A is correct. The standard form for the equation of a circle is , where
are the coordinates of the center and r is the length of the radius. According to the given equation, the center of the circle is
. Let
represent the coordinates of point Q. Since point P
and point Q
are the endpoints of a diameter of the circle, the center
lies on the diameter, halfway between P and Q. Therefore, the following relationships hold:
and
. Solving the equations for
and
, respectively, yields
and
. Therefore, the coordinates of point Q are
.
Alternate approach: Since point P on the circle and the center of the circle
have the same y-coordinate, it follows that the radius of the circle is
. In addition, the opposite end of the diameter
must have the same y-coordinate as P and be 4 units away from the center. Hence, the coordinates of point Q must be
.
Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter . If either of these points were point Q, then
would not be the diameter of the circle. Choice C is incorrect because
is the center of the circle and does not lie on the circle.
Question 196 196 of 269 selected Circles
A circle in the xy-plane has its center at and has a radius of . An equation of this circle is , where , , and are constants. What is the value of ?
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Correct Answer: -52The correct answer is . The equation of a circle in the xy-plane with its center at and a radius of can be written in the form . It's given that a circle in the xy-plane has its center at and has a radius of . Substituting for , for , and for in the equation yields , or . It's also given that an equation of this circle is , where , , and are constants. Therefore, can be rewritten in the form . The equation , or , can be rewritten as . Combining like terms on the left-hand side of this equation yields . Subtracting from both sides of this equation yields , which is equivalent to . This equation is in the form . Therefore, the value of is .
Question 197 197 of 269 selected Circles
In the xy-plane, a circle has center with coordinates . Points and lie on the circle. Point has coordinates , and is a right angle. What is the length of ?
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Correct Answer: AChoice A is correct. It's given that points and lie on the circle with center . Therefore, and are both radii of the circle. Since all radii of a circle are congruent, is congruent to . The length of , or the distance from point to point , can be found using the distance formula, which gives the distance between two points, and , as . Substituting the given coordinates of point , , for and the given coordinates of point , , for in the distance formula yields , or , which is equivalent to , or . Therefore, the length of is and the length of is . It's given that angle is a right angle. Therefore, triangle is a right triangle with legs and and hypotenuse . By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . Substituting for and in this equation yields , or , which is equivalent to . Taking the positive square root of both sides of this equation yields . Therefore, the length of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This would be the length of if the length of were , not .
Choice D is incorrect and may result from conceptual or calculation errors.
Question 198 198 of 269 selected Circles
The graph of the given equation in the xy-plane is a circle. What is the length of the radius of this circle?
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Correct Answer: CChoice C is correct. The equation of a circle in the xy-plane can be written as , where the center of the circle is and the radius of the circle is . The graph of the given equation, , is a circle in the xy-plane. This equation can be written as , where , , and . Therefore, the radius of this circle is .
Choice A is incorrect. This is the y-coordinate of the center, not the radius, of the circle defined by the given equation.
Choice B is incorrect. This is the x-coordinate of the center, not the radius, of the circle defined by the given equation.
Choice D is incorrect. This is the value of the radius squared, not the radius, of the circle defined by the given equation.
Question 199 199 of 269 selected Right Triangles & Trigonometry
The length of a rectangle’s diagonal is , and the length of the rectangle’s shorter side is . What is the length of the rectangle’s longer side?
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Correct Answer: 12The correct answer is . The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. It's given that the length of the rectangle's diagonal is and the length of the rectangle's shorter side is . Thus, the length of the hypotenuse of the right triangle formed by the diagonal is and the length of one of the legs is . By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . Substituting for and for in this equation yields , or . Subtracting from both sides of this equation yields . Taking the square root of both sides of this equation yields , or . Since represents a length, which must be positive, the value of is . Thus, the length of the rectangle's longer side is .
Question 200 200 of 269 selected Right Triangles & Trigonometry
In the right triangle shown, which of the following is closest to the value of ?
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Correct Answer: BChoice B is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . It's given in the right triangle shown that the legs have lengths of and and the hypotenuse has a length of . Substituting for , for , and for in yields , or . Taking the square root of both sides of this equation yields . Since the length of a side of a triangle must be positive, the value of is , which is approximately equal to . Of the choices, is the closest to the value of .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 201 201 of 269 selected Circles
The equation represents circle A. Circle B is obtained by shifting circle A down units in the xy-plane. Which of the following equations represents circle B?
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Correct Answer: AChoice A is correct. The standard form of an equation of a circle in the xy-plane is , where the coordinates of the center of the circle are and the length of the radius of the circle is . The equation of circle A, , can be rewritten as . Therefore, the center of circle A is at and the length of the radius of circle A is . If circle A is shifted down units, the y-coordinate of its center will decrease by ; the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at , or , and its radius is . Substituting for , for , and for in the equation yields , or . Therefore, the equation represents circle B.
Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, units.
Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, units.
Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, units.
Question 202 202 of 269 selected Area & Volume
Two identical rectangular prisms each have a height of . The base of each prism is a square, and the surface area of each prism is . If the prisms are glued together along a square base, the resulting prism has a surface area of . What is the side length, in , of each square base?
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Correct Answer: BChoice B is correct. Let represent the side length, in , of each square base. If the two prisms are glued together along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus the area of the two square bases that are being glued together, which yields . It’s given that this resulting surface area is equal to , so . Subtracting from both sides of this equation yields . This equation can be rewritten by multiplying on the left-hand side by , which yields , or . Adding to both sides of this equation yields . Multiplying both sides of this equation by yields . The surface area , in , of each rectangular prism is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of each rectangular prism is and the side length of each square base is , it follows that the area of each square base is and the area of each lateral face is . Therefore, the surface area of each rectangular prism can be represented by the expression , or . Substituting this expression for in the equation yields . Subtracting and from both sides of this equation yields . Factoring from the right-hand side of this equation yields . Applying the zero product property, it follows that and . Adding to both sides of the equation yields . Dividing both sides of this equation by yields . Since a side length of a rectangular prism can’t be , the length of each square base is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 203 203 of 269 selected Circles
In the xy-plane above, points P, Q, R, and T lie on the circle with center O. The degree measures of angles and
are each 30°. What is the radian measure of angle
?
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Correct Answer: CChoice C is correct. Because points T, O, and P all lie on the x-axis, they form a line. Since the angles on a line add up to , and it’s given that angles POQ and ROT each measure
, it follows that the measure of angle QOR is
. Since the arc of a complete circle is
or
radians, a proportion can be set up to convert the measure of angle QOR from degrees to radians:
, where x is the radian measure of angle QOR. Multiplying each side of the proportion by
gives
. Solving for x gives
, or
.
Choice A is incorrect and may result from subtracting only angle POQ from to get a value of
and then finding the radian measure equivalent to that value. Choice B is incorrect and may result from a calculation error. Choice D is incorrect and may result from calculating the sum of the angle measures, in radians, of angles POQ and ROT.
Question 204 204 of 269 selected Lines, Angles, & Triangles
In the figure, intersects at point , and is parallel to . The lengths of , , and are , , and , respectively. What is the length of ?
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Correct Answer: DChoice D is correct. The figure shows that angle and angle are vertical angles. Since vertical angles are congruent, angle and angle are congruent. It’s given that is parallel to . The figure also shows that intersects and . If two parallel segments are intersected by a third segment, alternate interior angles are congruent. Thus, alternate interior angles and are congruent. Since triangles and have two pairs of congruent angles, the triangles are similar. Sides and in triangle correspond to sides and , respectively, in triangle . Since the lengths of corresponding sides in similar triangles are proportional, it follows that . It's given that the lengths of , , and are , , and , respectively. Substituting for , for , and for in the equation yields . Multiplying each side of this equation by yields , or . It's given that intersects at point , so . Substituting for and for in this equation yields , or . Therefore, the length of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the length of , not .
Choice C is incorrect and may result from conceptual or calculation errors.
Question 205 205 of 269 selected Lines, Angles, & Triangles
In the figure above, segments AE and BD are parallel. If angle BDC measures 58° and angle ACE measures 62°, what is the measure of angle CAE ?
58°
60°
62°
120°
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Correct Answer: BChoice B is correct. It’s given that angle ACE measures . Since segments AE and BD are parallel, angles BDC and CEA are congruent. Therefore, angle CEA measures
. The sum of the measures of angles ACE, CEA, and CAE is
since the sum of the interior angles of triangle ACE is equal to
. Let the measure of angle CAE be
. Therefore,
, which simplifies to
. Thus, the measure of angle CAE is
.
Choice A is incorrect. This is the measure of angle AEC, not that of angle CAE. Choice C is incorrect. This is the measure of angle ACE, not that of CAE. Choice D is incorrect. This is the sum of the measures of angles ACE and CEA.
Question 206 206 of 269 selected Circles
The circle shown has center , circumference , and diameters and . The length of arc is twice the length of arc . What is the length of arc ?
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Correct Answer: BChoice B is correct. Since and are diameters of the circle shown, , , , and are radii of the circle and are therefore congruent. Since and are vertical angles, they are congruent. Therefore, arc and arc are formed by congruent radii and have the same angle measure, so they are congruent arcs. Similarly, and are vertical angles, so they are congruent. Therefore, arc and arc are formed by congruent radii and have the same angle measure, so they are congruent arcs. Let represent the length of arc . Since arc and arc are congruent arcs, the length of arc can also be represented by . It’s given that the length of arc is twice the length of arc . Therefore, the length of arc can be represented by the expression . Since arc and arc are congruent arcs, the length of arc can also be represented by . This gives the expression . Since it's given that the circumference is , the expression is equal to . Thus , or . Dividing both sides of this equation by yields . Therefore, the length of arc is , or .
Choice A is incorrect. This is the length of arc , not arc .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 207 207 of 269 selected Lines, Angles, & Triangles
In convex pentagon , segment is parallel to segment . The measure of angle is degrees, and the measure of angle is degrees. What is the measure, in degrees, of angle ?
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Correct Answer: 47The correct answer is . It's given that the measure of angle is degrees. Therefore, the exterior angle formed by extending segment at point has measure , or , degrees. It's given that segment is parallel to segment . Extending segment at point and extending segment at point until the two segments intersect results in a transversal that intersects two parallel line segments. One of these intersection points is point , and let the other intersection point be point . Since segment is parallel to segment , alternate interior angles are congruent. Angle and the exterior angle formed by extending segment at point are alternate interior angles. Therefore, the measure of angle is degrees. It's given that the measure of angle in pentagon is degrees. Therefore, angle has measure , or , degrees. Since angle in pentagon is an exterior angle of triangle , it follows that the measure of angle is the sum of the measures of angles and . Therefore, the measure, in degrees, of angle is , or .
Alternate approach: A line can be created that's perpendicular to segments and and passes through point . Extending segments and at points and , respectively, until they intersect this line yields two right triangles. Let these intersection points be point and point , and the two right triangles be triangle and triangle . It's given that the measure of angle is degrees. Therefore, angle has measure , or , degrees. Since the measure of angle is degrees and the measure of angle is degrees, it follows that the measure of angle is , or , degrees. It's given that the measure of angle is degrees. Therefore, angle has measure , or , degrees. Since the measure of angle is degrees and the measure of angle is degrees, it follows that the measure of angle is , or , degrees. Since angles , , and angle in pentagon form segment , it follows that the sum of the measures of those angles is degrees. Therefore, the measure, in degrees, of angle is , or .
Question 208 208 of 269 selected Area & Volume
The volume of right circular cylinder A is 22 cubic centimeters. What is the volume, in cubic centimeters, of a right circular cylinder with twice the radius and half the height of cylinder A?
11
22
44
66
Show Answer
Correct Answer: CChoice C is correct. The volume of right circular cylinder A is given by the expression , where r is the radius of its circular base and h is its height. The volume of a cylinder with twice the radius and half the height of cylinder A is given by
, which is equivalent to
. Therefore, the volume is twice the volume of cylinder A, or
.
Choice A is incorrect and likely results from not multiplying the radius of cylinder A by 2. Choice B is incorrect and likely results from not squaring the 2 in 2r when applying the volume formula. Choice D is incorrect and likely results from a conceptual error.
Question 209 209 of 269 selected Right Triangles & Trigonometry
Which of the following expressions is equivalent to ?
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Correct Answer: CChoice C is correct. The sine of an angle is equal to the cosine of its complementary angle. Since angles with measures and are complementary to each other, is equal to and is equal to . Substituting for and for in the given expression yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 210 210 of 269 selected Lines, Angles, & Triangles
In the figure, line intersects lines and . Line is parallel to line . What is the value of ?
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Correct Answer: 162The correct answer is . It's given that line is parallel to line . Since line intersects both lines and , it's a transversal. The angles in the figure marked as and are on the same side of the transversal, where one is an interior angle with line as a side, and the other is an exterior angle with line as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that the value of is .
Question 211 211 of 269 selected Right Triangles & Trigonometry
For the right triangle shown, and . Which expression represents the value of ?
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Correct Answer: DChoice D is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . In the right triangle shown, the hypotenuse has length and the legs have lengths and . It's given that and . Substituting for and for in the Pythagorean theorem yields . Taking the square root of both sides of this equation yields . Since the length of a side of a triangle must be positive, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 212 212 of 269 selected Area & Volume
The area of a rectangle is square inches. The length of the rectangle is inches. What is the width, in inches, of this rectangle?
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Correct Answer: AChoice A is correct. The area , in square inches, of a rectangle is the product of its length , in inches, and its width , in inches; thus, . It's given that the area of a rectangle is square inches and the length of the rectangle is inches. Substituting for and for in the equation yields . Dividing both sides of this equation by yields . Therefore, the width, in inches, of this rectangle is .
Choice B is incorrect. This is the length, not the width, in inches, of the rectangle.
Choice C is incorrect. This is half the area, in square inches, not the width, in inches, of the rectangle.
Choice D is incorrect. This is the difference between the area, in square inches, and the length, in inches, of the rectangle, not the width, in inches, of the rectangle.
Question 213 213 of 269 selected Area & Volume
What is the area of a rectangle with a length of and a width of ?
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Correct Answer: CChoice C is correct. The area of a rectangle with length and width can be found using the formula . It’s given that the rectangle has a length of and a width of . Therefore, the area of this rectangle is , or .
Choice A is incorrect. This is the sum of the length and width of the rectangle, not the area.
Choice B is incorrect. This is the perimeter of the rectangle, not the area.
Choice D is incorrect. This is the sum of the length and width of the rectangle squared, not the area.
Question 214 214 of 269 selected Right Triangles & Trigonometry
In the triangle shown, what is the value of ?
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Correct Answer: 45The correct answer is . An isosceles right triangle has a right angle and two legs of equal length. In the triangle shown, one angle is a right angle and the two legs each have a length of . Thus, the given triangle is an isosceles right triangle. In an isosceles right triangle, the measures of the two non-right angles are . It follows that the value of is .
Question 215 215 of 269 selected Right Triangles & Trigonometry
The length of a rectangle’s diagonal is , and the length of the rectangle’s shorter side is . What is the length of the rectangle’s longer side?
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Correct Answer: BChoice B is correct. A rectangle’s diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It’s given that the length of the diagonal is and the length of the rectangle’s shorter side is . Therefore, each of the two right triangles formed by the rectangle’s diagonal has a hypotenuse with length , and a shorter leg with length . To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, , can be used, where and are the lengths of the legs and is the length of the hypotenuse of the triangle. Substituting for and for in the equation yields , which is equivalent to , or . Subtracting from each side of this equation yields . Taking the positive square root of each side of this equation yields . Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is . It follows that the length of the rectangle’s longer side is .
Choice A is incorrect and may result from dividing the length of the rectangle’s diagonal by the length of the rectangle’s shorter side, rather than substituting these values into the Pythagorean theorem.
Choice C is incorrect and may result from using the length of the rectangle’s diagonal as the length of a leg of the right triangle, rather than the length of the hypotenuse.
Choice D is incorrect. This is the square of the length of the rectangle’s longer side.
Question 216 216 of 269 selected Area & Volume
The area of a rectangle is square inches. The length of the longest side of the rectangle is inches. What is the length, in inches, of the shortest side of this rectangle?
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Correct Answer: 3The correct answer is . The area of a rectangle can be calculated by multiplying the length of its longest side by the length of its shortest side. It’s given that the area of the rectangle is square inches and the length of the longest side of the rectangle is inches. Let represent the length, in inches, of the shortest side of this rectangle. It follows that . Dividing both sides of this equation by yields . Therefore, the length, in inches, of the shortest side of the rectangle is .
Question 217 217 of 269 selected Circles
Points A and B lie on a circle with radius 1, and arc has length
. What fraction of the circumference of the circle is the length of arc
?
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The correct answer is . The circumference, C, of a circle is
, where r is the length of the radius of the circle. For the given circle with a radius of 1, the circumference is
, or
. To find what fraction of the circumference the length of arc
is, divide the length of the arc by the circumference, which gives
. This division can be represented by
. Note that 1/6, .1666, .1667, 0.166, and 0.167 are examples of ways to enter a correct answer.
Question 218 218 of 269 selected Right Triangles & Trigonometry
A square is inscribed in a circle. The radius of the circle is inches. What is the side length, in inches, of the square?
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Correct Answer: AChoice A is correct. When a square is inscribed in a circle, a diagonal of the square is a diameter of the circle. It's given that a square is inscribed in a circle and the length of a radius of the circle is inches. Therefore, the length of a diameter of the circle is inches, or inches. It follows that the length of a diagonal of the square is inches. A diagonal of a square separates the square into two right triangles in which the legs are the sides of the square and the hypotenuse is a diagonal. Since a square has congruent sides, each of these two right triangles has congruent legs and a hypotenuse of length inches. Since each of these two right triangles has congruent legs, they are both -- triangles. In a -- triangle, the length of the hypotenuse is times the length of a leg. Let represent the length of a leg of one of these -- triangles. It follows that . Dividing both sides of this equation by yields . Therefore, the length of a leg of one of these -- triangles is inches. Since the legs of these two -- triangles are the sides of the square, it follows that the side length of the square is inches.
Choice B is incorrect. This is the length of a radius, in inches, of the circle.
Choice C is incorrect. This is the length of a diameter, in inches, of the circle.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 219 219 of 269 selected Right Triangles & Trigonometry
Triangle is similar to triangle , where angle corresponds to angle and angles and are right angles. The length of is times the length of . If , what is the value of ?
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Correct Answer: .7241, 21/29The correct answer is . It's given that triangle is similar to triangle , where angle corresponds to angle and angles and are right angles. In similar triangles, the tangents of corresponding angles are equal. Therefore, if , then . In a right triangle, the tangent of an acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, in triangle , if , the ratio of the length of to the length of is . If the lengths of and are and , respectively, then the ratio of the length of to the length of is . In a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, the value of is the ratio of the length of to the length of . The length of can be calculated using the Pythagorean theorem, which states that if the lengths of the legs of a right triangle are and and the length of the hypotenuse is , then . Therefore, if the lengths of and are and , respectively, then , or . Taking the positive square root of both sides of this equation yields . Therefore, if the lengths of and are and , respectively, then the length of is and the ratio of the length of to the length of is . Thus, if , the value of is . Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.
Question 220 220 of 269 selected Area & Volume
What is the area, in square units, of the triangle formed by connecting the three points shown?
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Correct Answer: 24.5, 49/2The correct answer is . It's given that a triangle is formed by connecting the three points shown, which are , , and . Let this triangle be triangle A. The area of triangle A can be found by calculating the area of the rectangle that circumscribes it and subtracting the areas of the three triangles that are inside the rectangle but outside triangle A. The rectangle formed by the points , , , and circumscribes triangle A. The width, in units, of this rectangle can be found by calculating the distance between the points and . This distance is , or . The length, in units, of this rectangle can be found by calculating the distance between the points and . This distance is , or . It follows that the area, in square units, of the rectangle is , or . One of the triangles that lies inside the rectangle but outside triangle A is formed by the points , , and . The length, in units, of a base of this triangle can be found by calculating the distance between the points and . This distance is , or . The corresponding height, in units, of this triangle can be found by calculating the distance between the points and . This distance is , or . It follows that the area, in square units, of this triangle is , or . A second triangle that lies inside the rectangle but outside triangle A is formed by the points , , and . The length, in units, of a base of this triangle can be found by calculating the distance between the points and . This distance is , or . The corresponding height, in units, of this triangle can be found by calculating the distance between the points and . This distance is , or . It follows that the area, in square units, of this triangle is , or . The third triangle that lies inside the rectangle but outside triangle A is formed by the points , , and . The length, in units, of a base of this triangle can be found by calculating the distance between the points and . This distance is , or . The corresponding height, in units, of this triangle can be found by calculating the distance between the points and . This distance is , or . It follows that the area, in square units, of this triangle is , or . Thus, the area, in square units, of the triangle formed by connecting the three points shown is , or . Note that 24.5 and 49/2 are examples of ways to enter a correct answer.
Question 221 221 of 269 selected Lines, Angles, & Triangles
In the figure, two lines intersect at a point. If , what is the value of ?
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Correct Answer: DChoice D is correct. In the figure shown, the angles with measures and are vertical angles. Since vertical angles are congruent, . Therefore, if , the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the measure, in degrees, of an angle that's supplementary, not congruent, to the angle with measure .
Choice C is incorrect and may result from conceptual or calculation errors.
Question 222 222 of 269 selected Area & Volume
Circle has a radius of and circle has a radius of , where is a positive constant. The area of circle is how many times the area of circle ?
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Correct Answer: DChoice D is correct. The area of a circle can be found by using the formula , where is the area and is the radius of the circle. It’s given that the radius of circle A is . Substituting this value for into the formula gives , or . It’s also given that the radius of circle B is . Substituting this value for into the formula gives , or . Dividing the area of circle B by the area of circle A gives , which simplifies to . Therefore, the area of circle B is times the area of circle A.
Choice A is incorrect. This is how many times greater the radius of circle B is than the radius of circle A.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the coefficient on the term that describes the radius of circle B.
Question 223 223 of 269 selected Area & Volume
A triangle has a base length of centimeters and a height of centimeters. What is the area, in square centimeters, of the triangle?
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Correct Answer: 1800The correct answer is . The area, , of a triangle can be found using the formula , where is the base length of the triangle and is the height of the triangle. It’s given that the triangle has a base length of centimeters and a height of centimeters. Substituting for and for in the formula yields , or . Therefore, the area, in square centimeters, of the triangle is .
Question 224 224 of 269 selected Right Triangles & Trigonometry
In a right triangle, the tangent of one of the two acute angles is . What is the tangent of the other acute angle?
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Correct Answer: DChoice D is correct. The tangent of a nonright angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Using that definition for tangent, in a right triangle with legs that have lengths a and b, the tangent of one acute angle is and the tangent for the other acute angle is
. It follows that the tangents of the acute angles in a right triangle are reciprocals of each other. Therefore, the tangent of the other acute angle in the given triangle is the reciprocal of
or
.
Choice A is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the tangent of the angle described. Choice B is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the reciprocal of the tangent of the angle described. Choice C is incorrect and may result from interpreting the tangent of the other acute angle as equal to the tangent of the angle described.
Question 225 225 of 269 selected Area & Volume
A right circular cylinder has a base diameter of centimeters and a height of centimeters. What is the volume, in cubic centimeters, of the cylinder?
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Correct Answer: CChoice C is correct. The volume, , of a right circular cylinder is given by the formula , where is the radius of the base of the cylinder and is the height of the cylinder. It's given that a right circular cylinder has a height of centimeters. Therefore, . It's also given that the cylinder has a base diameter of centimeters. The radius of a circle is half the diameter of the circle. Since the base of a right circular cylinder is a circle, it follows that the radius of the base of the right circular cylinder is , or , centimeters. Therefore, . Substituting for and for in the formula yields , which is equivalent to , or . Therefore, the volume, in cubic centimeters, of the cylinder is .
Choice A is incorrect. This is the volume of a right circular cylinder that has a base diameter of , not , centimeters and a height of centimeters.
Choice B is incorrect. This is the volume of a right circular cylinder that has a base diameter of , not , centimeters and a height of centimeters.
Choice D is incorrect. This is the volume of a right circular cylinder that has a base diameter of , not , centimeters and a height of centimeters.
Question 226 226 of 269 selected Right Triangles & Trigonometry
In the right triangle shown above, what is the length of ?
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The correct answer is 4. Triangle PQR has given angle measures of 30° and 90°, so the third angle must be 60° because the measures of the angles of a triangle sum to 180°. For any special right triangle with angles measuring 30°, 60°, and 90°, the length of the hypotenuse (the side opposite the right angle) is 2x, where x is the length of the side opposite the 30° angle. Segment PQ is opposite the 30° angle. Therefore, 2(PQ) = 8 and PQ = 4.
Question 227 227 of 269 selected Area & Volume
Each side of a square has a length of . What is the perimeter of this square?
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Correct Answer: 180The correct answer is . The perimeter of a polygon is equal to the sum of the lengths of the sides of the polygon. It’s given that each side of the square has a length of . Since a square is a polygon with sides, the perimeter of this square is , or .
Question 228 228 of 269 selected Lines, Angles, & Triangles
In the figure, parallel lines and are intersected by lines and . If and , what is the value of ?
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Correct Answer: 101/2, 50.5The correct answer is . In the figure, lines , , and form a triangle. One interior angle of this triangle is vertical to the angle marked ; therefore, the interior angle also has measure . It's given that . Therefore, the interior angle of the triangle has measure . A second interior angle of the triangle forms a straight line, , with the angle marked . Therefore, the sum of the measures of these two angles is . It's given that . Therefore, the angle marked has measure and the second interior angle of the triangle has measure , or . The sum of the interior angles of a triangle is . Therefore, the measure of the third interior angle of the triangle is , or . It's given that parallel lines and are intersected by line . It follows that the triangle's interior angle with measure is congruent to the same side interior angle between lines and formed by lines and . Since this angle is supplementary to the two angles marked , the sum of , , and is . It follows that , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Note that 101/2 and 50.5 are examples of ways to enter a correct answer.
Question 229 229 of 269 selected Area & Volume
What is the area of a rectangle with a length of and a width of ?
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Correct Answer: BChoice B is correct. The area of a rectangle with length and width can be found using the formula . It’s given that the rectangle has a length of and a width of . Therefore, the area of this rectangle is , or .
Choice A is incorrect. This is the sum, , of the length and width of the rectangle, not the area, .
Choice C is incorrect. This is the perimeter, , of the rectangle, not the area, .
Choice D is incorrect. This is the sum of the length and width of the rectangle squared, not the area.
Question 230 230 of 269 selected Area & Volume
The line segment shown in the xy-plane represents one of the legs of a right triangle. The area of this triangle is square units. What is the length, in units, of the other leg of this triangle?
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Correct Answer: BChoice B is correct. The length of a segment in the xy-plane can be found using the distance formula, , where and are the endpoints of the segment. The segment shown has endpoints at and . Substituting and for and , respectively, in the distance formula yields , or , which is equivalent to , or . Let represent the length, in units, of the other leg of this triangle. The area, , of a right triangle can be calculated using the formula , where and are the lengths of the legs of the triangle. It's given that the area of the triangle is square units. Substituting for , for , and for in the formula yields . Multiplying both sides of this equation by yields . Dividing both sides of this equation by yields . Multiplying the numerator and denominator on the left-hand side of this equation by yields , or , which is equivalent to , or . Therefore, the length, in units, of the other leg of this triangle is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. is equivalent to , which is the length, in units, of the line segment shown in the xy-plane, not the length, in units, of the other leg of the triangle.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 231 231 of 269 selected Right Triangles & Trigonometry
In the triangle shown, . What is the value of ?
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Correct Answer: .9811, 52/53The correct answer is . In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. The length of the hypotenuse of the right triangle shown is . It’s given that . Therefore, the length of one of the legs of the triangle shown is . Let represent , the length of the other leg of the triangle shown. Therefore, , or . Subtracting from both sides of this equation yields . Taking the positive square root of both sides of this equation yields . Therefore, , the length of the other leg of the triangle shown, is . The sine of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The length of the leg opposite angle is , and the length of the hypotenuse is . Therefore, the value of is . Note that 52/53 or .9811 are examples of ways to enter a correct answer.
Question 232 232 of 269 selected Area & Volume
A rectangle has a length of inches and a width of inches. What is the area, in square inches, of the rectangle?
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Correct Answer: 2048The correct answer is . The area , in square inches, of a rectangle is equal to the product of its length , in inches, and its width , in inches, or . It's given that the rectangle has a length of inches and a width of inches. Substituting for and for in the equation yields , or . Therefore, the area, in square inches, of the rectangle is .
Question 233 233 of 269 selected Lines, Angles, & Triangles
In the figure shown, units, units, and units. What is the area, in square units, of triangle ?
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Correct Answer: 480The correct answer is . It's given in the figure that angle and angle are right angles. It follows that angle is congruent to angle . It's also given that angle and angle are the same angle. It follows that angle is congruent to angle . Since triangles and have two pairs of congruent angles, the triangles are similar. Sides and in triangle correspond to sides and , respectively, in triangle . Corresponding sides in similar triangles are proportional. Therefore, . It's given that units and units. Therefore, units. It’s also given that units. Substituting for , for , and for in the equation yields , or . Multiplying each side of this equation by yields . By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then . Since triangle is a right triangle, it follows that represents the length of the hypotenuse, , and and represent the lengths of the legs, and . Substituting for and for in the equation yields , which is equivalent to , or . Subtracting from both sides of this equation yields . Taking the square root of both sides of this equation yields . Since represents a length, which must be positive, the value of is . Therefore, . Since and represent the lengths of the legs of triangle , it follows that and can be used to calculate the area, in square units, of the triangle as , or . Therefore, the area, in square units, of triangle is .
Question 234 234 of 269 selected Area & Volume
The dimensions of a right rectangular prism are 4 inches by 5 inches by 6 inches. What is the surface area, in square inches, of the prism?
30
74
120
148
Show Answer
Correct Answer: DChoice D is correct. The surface area is found by summing the area of each face. A right rectangular prism consists of three pairs of congruent rectangles, so the surface area is found by multiplying the areas of three adjacent rectangles by 2 and adding these products. For this prism, the surface area is equal to , or
, which is equal to 148.
Choice A is incorrect. This is the area of one of the faces of the prism. Choice B is incorrect and may result from adding the areas of three adjacent rectangles without multiplying by 2. Choice C is incorrect. This is the volume, in cubic inches, of the prism.
Question 235 235 of 269 selected Right Triangles & Trigonometry
Triangle is similar to triangle , where angle corresponds to angle and angle corresponds to angle . Angles and are right angles. If , what is the value of ?
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Correct Answer: .14, 7/50The correct answer is . It's given that triangle is similar to triangle , where angle corresponds to angle and angle corresponds to angle . In similar triangles, the tangents of corresponding angles are equal. Since angle and angle are corresponding angles, if , then . It's also given that angles and are right angles. It follows that triangle is a right triangle with acute angles and . The tangent of one acute angle in a right triangle is the inverse of the tangent of the other acute angle in the triangle. Therefore, . Substituting for in this equation yields , or . Thus, if , the value of is . Note that 7/50 and .14 are examples of ways to enter a correct answer.
Question 236 236 of 269 selected Lines, Angles, & Triangles
In the figure above, lines m and n are parallel. What is the value of b ?
40
50
65
80
Show Answer
Correct Answer: AChoice A is correct. Given that lines m and n are parallel, the angle marked 130° must be supplementary to the leftmost angle marked a° because they are same-side interior angles. Therefore, 130° + a° = 180°, which yields a = 50°. Lines and m intersect at a right angle, so lines j,
, and m form a right triangle where the two acute angles are a° and b°. The acute angles of a right triangle are complementary, so a° + b° = 90°, which yields 50° + b° = 90°, and b = 40.
Choice B is incorrect. This is the value of a, not b. Choice C is incorrect and may be the result of dividing 130° by 2. Choice D is incorrect and may be the result of multiplying b by 2.
Question 237 237 of 269 selected Circles
Circle A (shown) is defined by the equation . Circle B (not shown) is the result of shifting circle A down units and increasing the radius so that the radius of circle B is times the radius of circle A. Which equation defines circle B?
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Correct Answer: AChoice A is correct. According to the graph, the center of circle A has coordinates , and the radius of circle A is . It’s given that circle B is the result of shifting circle A down units and increasing the radius so that the radius of circle B is times the radius of circle A. It follows that the center of circle B is units below the center of circle A. The point that's units below has the same x-coordinate as and has a y-coordinate that is less than the y-coordinate of . Therefore, the coordinates of the center of circle B are , or . Since the radius of circle B is times the radius of circle A, the radius of circle B is . A circle in the xy-plane can be defined by an equation of the form , where the coordinates of the center of the circle are and the radius of the circle is . Substituting for , for , and for in this equation yields , which is equivalent to , or . Therefore, the equation defines circle B.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This equation defines a circle that’s the result of shifting circle A up, not down, by units and increasing the radius.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 238 238 of 269 selected Lines, Angles, & Triangles
In triangle , and . Which statement is sufficient to prove that triangle is equilateral?
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Correct Answer: AChoice A is correct. In an equilateral triangle, all three sides have the same length. It’s given that in triangle , and . Therefore, if , then all three sides of triangle have the same length, so triangle is equilateral. Therefore, is sufficient to prove that triangle is equilateral.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 239 239 of 269 selected Right Triangles & Trigonometry
In the figure above, is parallel to
. What is the length of
?
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The correct answer is 30. In the figure given, since is parallel to
and both segments are intersected by
, then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE,
corresponds to
and
corresponds to
. Therefore,
. Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD:
. Taking the square root of each side gives
. Substituting the values in the proportion
yields
. Multiplying each side by CE, and then multiplying by
yields
. Therefore, the length of
is 30.
Question 240 240 of 269 selected Area & Volume
A hemisphere is half of a sphere. If a hemisphere has a radius of inches, which of the following is closest to the volume, in cubic inches, of this hemisphere?
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Correct Answer: DChoice D is correct. The volume, , of a sphere is given by , where is the radius of the sphere. Since a hemisphere is half of a sphere, it follows that the volume, , of a hemisphere is given by , or . Substituting for in this formula yields , which gives , or is approximately equal to . Therefore, the choice that is closest to the volume, in cubic inches, of this hemisphere is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 241 241 of 269 selected Circles
A circle in the xy-plane has equation . Which of the following points does NOT lie in the interior of the circle?
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Correct Answer: DChoice D is correct. The circle with equation has center
and radius 5. For a point to be inside of the circle, the distance from that point to the center must be less than the radius, 5. The distance between
and
is
, which is greater than 5. Therefore,
does NOT lie in the interior of the circle.
Choice A is incorrect. The distance between and
is
, which is less than 5, and therefore
lies in the interior of the circle. Choice B is incorrect because it is the center of the circle. Choice C is incorrect because the distance between
and
is
, which is less than 5, and therefore
in the interior of the circle.
Question 242 242 of 269 selected Lines, Angles, & Triangles
Right triangles and are similar, where and correspond to and , respectively. Angle has a measure of . What is the measure of angle ?
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Correct Answer: BChoice B is correct. It’s given that triangle is similar to triangle . Corresponding angles of similar triangles are congruent. Since angle and angle correspond to each other, they must be congruent. Therefore, if the measure of angle is , then the measure of angle is also .
Choice A is incorrect and may result from concluding that angle and angle are complementary rather than congruent.
Choice C is incorrect and may result from concluding that angle and angle are supplementary rather than congruent.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 243 243 of 269 selected Right Triangles & Trigonometry
In triangle , and angle is a right angle. What is the value of ?
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Correct Answer: .8823, .8824, 15/17The correct answer is . It's given that angle is the right angle in triangle . Therefore, the acute angles of triangle are angle and angle . The hypotenuse of a right triangle is the side opposite its right angle. Therefore, the hypotenuse of triangle is side . The cosine of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. It's given that . This can be written as . Since the cosine of angle is a ratio, it follows that the length of the side adjacent to angle is and the length of the hypotenuse is , where is a constant. Therefore, and . The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For triangle , it follows that . Substituting for and for yields . This is equivalent to . Subtracting from each side of this equation yields . Taking the square root of each side of this equation yields . Since , it follows that , which can be rewritten as . Note that 15/17, .8824, .8823, and 0.882 are examples of ways to enter a correct answer.
Question 244 244 of 269 selected Lines, Angles, & Triangles
For the triangles shown, triangle is dilated by a scale factor of to obtain triangle , where . What is the measure, in degrees, of angle ?
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Correct Answer: CChoice C is correct. It's given that triangle is obtained by a dilation of triangle . It follows that triangle is similar to triangle , where corresponds to . Since corresponding angles in similar triangles have the same measure and the measure of angle is degrees, it follows that the measure of angle is also degrees.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Question 245 245 of 269 selected Area & Volume
A right circular cylinder has a volume of cubic centimeters. The area of the base of the cylinder is square centimeters. What is the height, in centimeters, of the cylinder?
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Correct Answer: AChoice A is correct. The volume, , of a right circular cylinder is given by the formula , where is the area of the base of the cylinder and is the height. It’s given that a right circular cylinder has a volume of cubic centimeters and the area of the base is square centimeters. Substituting for and for in the formula yields . Dividing both sides of this equation by yields . Therefore, the height of the cylinder, in centimeters, is .
Choice B is incorrect. This is the area of the base, in square centimeters, not the height, in centimeters, of the cylinder.
Choice C is incorrect. This is the height, in centimeters, of a cylinder if its volume is cubic centimeters and the area of its base is , not , cubic centimeters.
Choice D is incorrect. This is the height, in centimeters, of a cylinder if its volume is cubic centimeters and the area of its base is , not , cubic centimeters.
Question 246 246 of 269 selected Lines, Angles, & Triangles
In a right triangle, the measure of one of the acute angles is . What is the measure, in degrees, of the other acute angle?
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Correct Answer: BChoice B is correct. The sum of the measures of the interior angles of a triangle is degrees. Since the triangle is a right triangle, it has one angle that measures degrees. Therefore, the sum of the measures, in degrees, of the remaining two angles is , or . It’s given that the measure of one of the acute angles in the triangle is degrees. Therefore, the measure, in degrees, of the other acute angle is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the measure, in degrees, of the acute angle whose measure is given.
Question 247 247 of 269 selected Area & Volume
The figure shows the lengths, in centimeters (cm), of the edges of a right rectangular prism. The volume V of a right rectangular prism is , where
is the length of the prism, w is the width of the prism, and h is the height of the prism. What is the volume, in cubic centimeters, of the prism?
36
24
12
11
Show Answer
Correct Answer: AChoice A is correct. It’s given that the volume of a right rectangular prism is . The prism shown has a length of 6 cm, a width of 2 cm, and a height of 3 cm. Thus,
, or 36 cubic centimeters.
Choice B is incorrect. This is the volume of a rectangular prism with edge lengths of 6, 2, and 2. Choice C is incorrect and may result from only finding the product of the length and width of the base of the prism. Choice D is incorrect and may result from finding the sum, not the product, of the edge lengths of the prism.
Question 248 248 of 269 selected Right Triangles & Trigonometry
In the triangle shown, what is the value of ?
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Correct Answer: .3928, .3929, 11/28The correct answer is . The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to the angle with measure is units and the length of the hypotenuse is units. Therefore, the value of is . Note that 11/28, .3928, .3929, 0.392, and 0.393 are examples of ways to enter a correct answer.
Question 249 249 of 269 selected Right Triangles & Trigonometry
For two acute angles, and , . The measures, in degrees, of and are and , respectively. What is the value of ?
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Correct Answer: AChoice A is correct. It's given that for two acute angles, and , . For two acute angles, if the sine of one angle is equal to the cosine of the other angle, the angles are complementary. It follows that and are complementary. That is, the sum of the measures of the angles is degrees. It's given that the measure of is degrees and the measure of is degrees. It follows that . By combining like terms, this equation can be rewritten as . Subtracting from each side of this equation yields . Dividing each side of this equation by yields .
Choice B is incorrect. This would be the value of if rather than .
Choice C is incorrect. This would be the value of if rather than and if were obtuse rather than acute.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 250 250 of 269 selected Area & Volume
A cylindrical can containing pieces of fruit is filled to the top with syrup before being sealed. The base of the can has an area of , and the height of the can is 10 cm. If
of syrup is needed to fill the can to the top, which of the following is closest to the total volume of the pieces of fruit in the can?
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Correct Answer: CChoice C is correct. The total volume of the cylindrical can is found by multiplying the area of the base of the can, , by the height of the can, 10 cm, which yields
. If the syrup needed to fill the can has a volume of
, then the remaining volume for the pieces of
fruit is .
Choice A is incorrect because if the fruit had a volume of , there would be
of syrup needed to fill the can to the top. Choice B is incorrect because if the fruit had a volume of
, there would be
of syrup needed to fill the can to the top. Choice D is incorrect because it is the total volume of the can, not just of the pieces of fruit.
Question 251 251 of 269 selected Area & Volume
A triangle has a base length of centimeters and a height of centimeters. What is the area, in square centimeters, of the triangle?
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Correct Answer: CChoice C is correct. The area, , of a triangle is given by the formula , where is the base length and is the height of the triangle. It’s given that a triangle has a base length of centimeters and a height of centimeters. Substituting for and for in the formula yields , or . Therefore, the area, in square centimeters, of the triangle is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
Question 252 252 of 269 selected Right Triangles & Trigonometry
In the triangle shown, what is the value of ?
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Correct Answer: AChoice A is correct. Since the two acute angles have the same measure and the third angle is a right angle, the triangle shown is an isosceles right triangle. In an isosceles right triangle, the two legs have the same length. The figure shows that the length of one leg of the triangle is and the length of the other leg of the triangle is . It follows that the value of is .
Choice B is incorrect. This is the measure, in degrees, of one of the angles shown.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Question 253 253 of 269 selected Lines, Angles, & Triangles
In the figure above, two sides of a triangle are extended. What is the value of x ?
110
120
130
140
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Correct Answer: BChoice B is correct. The sum of the interior angles of a triangle is 180°. The measures of the two interior angles of the given triangle are shown. Therefore, the measure of the third interior angle is 180° – 70° – 50° = 60°. The angles of measures x° and 60° are supplementary, so their sum is 180°. Therefore, x = 180 – 60 = 120.
Choice A is incorrect and may be the result of misinterpreting x° as supplementary to 70°. Choice C is incorrect and may be the result of misinterpreting x° as supplementary to 50°. Choice D is incorrect and may be the result of a calculation error.
Question 254 254 of 269 selected Area & Volume
The width of a rectangle is centimeters. The length of the rectangle is centimeters longer than the width. What is the area, in square centimeters, of this rectangle?
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Correct Answer: DChoice D is correct. It’s given that the width of this rectangle is centimeters and that the length of this rectangle is centimeters longer than the width. Therefore, the length of this rectangle is , or , centimeters. The area of a rectangle can be found by multiplying its length and its width. Therefore the area, in square centimeters, of this rectangle is , or .
Choice A is incorrect. This is the width, in centimeters, not the area, in square centimeters, of this rectangle.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Question 255 255 of 269 selected Lines, Angles, & Triangles
In triangle , the measure of angle is , the measure of angle is , and the measure of angle is . What is the value of ?
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Correct Answer: CChoice C is correct. The sum of the interior angles of a triangle is . It's given that the interior angles of triangle are , , and . It follows that , or . Subtracting from each side of this equation yields . Multiplying each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 256 256 of 269 selected Area & Volume
A right triangle has sides of length , , and units. What is the area of the triangle, in square units?
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Correct Answer: BChoice B is correct. The area, , of a triangle can be found using the formula , where is the length of the base of the triangle and is the height of the triangle. It's given that the triangle is a right triangle. Therefore, its base and height can be represented by the two legs. It’s also given that the triangle has sides of length , , and units. Since units is the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths and units. Substituting these values for and in the formula gives , which is equivalent to square units, or square units.
Choice A is incorrect. This expression represents the perimeter, rather than the area, of the triangle.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 257 257 of 269 selected Lines, Angles, & Triangles
In the figure above, and
. What is the length of
?
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The correct answer is 4.5. According to the properties of right triangles, BD divides triangle ABC into two similar triangles, ABD and BCD. The corresponding sides of ABD and BCD are proportional, so the ratio of BD to AD is the same as the ratio of DC to BD. Expressing this information as a proportion gives . Solving the proportion for DC results in
. Note that 4.5 and 9/2 are examples of ways to enter a correct answer.
Question 258 258 of 269 selected Area & Volume
The glass pictured above can hold a maximum volume of 473 cubic centimeters, which is approximately 16 fluid ounces. What is the value of k, in centimeters?
2.52
7.67
7.79
10.11
Show Answer
Correct Answer: DChoice D is correct. Using the volume formula and the given information that the volume of the glass is 473 cubic centimeters, the value of k can be found as follows:
Therefore, the value of k is approximately 10.11 centimeters.
Choices A, B, and C are incorrect. Substituting the values of k from these choices in the formula results in volumes of approximately 7 cubic centimeters, 207 cubic centimeters, and 217 cubic centimeters, respectively, all of which contradict the given information that the volume of the glass is 473 cubic centimeters.
Question 259 259 of 269 selected Lines, Angles, & Triangles
A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of the acute angles is . The sum of the measures of one of the acute angles and three of the obtuse angles is . What is the value of ?
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Correct Answer: 1660The correct answer is . It’s given that a line intersects two parallel lines, forming four acute angles and four obtuse angles. When two parallel lines are intersected by a transversal line, the angles formed have the following properties: two adjacent angles are supplementary, and alternate interior angles are congruent. Therefore, each of the four acute angles have the same measure, and each of the four obtuse angles have the same measure. It’s also given that the measure of one of the acute angles is . If two angles are supplementary, then the sum of their measures is . Therefore, the measure of the obtuse angle adjacent to any of the acute angles is , or , which is equivalent to . It’s given that the sum of the measures of one of the acute angles and three of the obtuse angles is . It follows that , which is equivalent to , or . Adding to both sides of this equation yields .
Question 260 260 of 269 selected Area & Volume
What is the perimeter, in inches, of a rectangle with a length of inches and a width of inches?
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Correct Answer: DChoice D is correct. The perimeter of a figure is equal to the sum of the measurements of the sides of the figure. It’s given that the rectangle has a length of inches and a width of inches. Since a rectangle has sides, of which opposite sides are parallel and equal, it follows that the rectangle has two sides with a length of inches and two sides with a width of inches. Therefore, the perimeter of this rectangle is , or inches.
Choice A is incorrect. This is the sum, in inches, of the length and the width of the rectangle.
Choice B is incorrect. This is the sum, in inches, of the two lengths and the width of the rectangle.
Choice C is incorrect. This is the sum, in inches, of the length and the two widths of the rectangle.
Question 261 261 of 269 selected Circles
A circle in the xy-plane has its center at and the point lies on the circle. Which equation represents this circle?
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Correct Answer: DChoice D is correct. A circle in the xy-plane can be represented by an equation of the form , where is the center of the circle and is the length of a radius of the circle. It's given that the circle has its center at . Therefore, and . Substituting for and for in the equation yields , or . It's also given that the point lies on the circle. Substituting for and for in the equation yields , or , which is equivalent to , or . Substituting for in the equation yields . Thus, the equation represents the circle.
Choice A is incorrect. The circle represented by this equation has its center at , not , and the point doesn't lie on the circle.
Choice B is incorrect. The point doesn't lie on the circle represented by this equation.
Choice C is incorrect. The circle represented by this equation has its center at , not , and the point doesn't lie on the circle.
Question 262 262 of 269 selected Area & Volume
Triangles and are similar. Each side length of triangle is times the corresponding side length of triangle . The area of triangle is square inches. What is the area, in square inches, of triangle ?
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Correct Answer: 135/8, 16.87, 16.88The correct answer is . It's given that triangles and are similar and each side length of triangle is times the corresponding side length of triangle . For two similar triangles, if each side length of the first triangle is times the corresponding side length of the second triangle, then the area of the first triangle is times the area of the second triangle. Therefore, the area of triangle is , or , times the area of triangle . It's given that the area of triangle is square inches. Let represent the area, in square inches, of triangle . It follows that is times , or . Dividing both sides of this equation by yields , which is equivalent to . Thus, the area, in square inches, of triangle is . Note that 135/8, 16.87, and 16.88 are examples of ways to enter a correct answer.
Question 263 263 of 269 selected Circles
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle's radius?
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Correct Answer: CChoice C is correct. It's given that in the xy-plane, the graph of the given equation is a circle. The equation of a circle in the xy-plane can be written in the form , where is the center of the circle and is the length of the circle's radius. Subtracting from both sides of the equation yields . By completing the square, this equation can be rewritten as . This equation can be rewritten as , or . Therefore, . Taking the square root of both sides of this equation yields and . Since is the length of the circle's radius, must be positive. Therefore, the length of the circle's radius is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question 264 264 of 269 selected Lines, Angles, & Triangles
In the figure, line is parallel to line , and line intersects both lines. What is the value of ?
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Correct Answer: DChoice D is correct. In the figure shown, the angle marked and the angle marked form a linear pair of angles. If two angles form a linear pair of angles, the sum of the measures of the angles is . Therefore, the value of is .
Choice A is incorrect. This is less than , not the sum of and .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the measure, in degrees, of one of the angles shown.
Question 265 265 of 269 selected Circles
What is the value of ?
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Correct Answer: AChoice A is correct. The sine of a number is the y-coordinate of the point arrived at by traveling a distance of units counterclockwise around the unit circle from the starting point . Since the unit circle has a circumference of units, it follows that one full rotation around the circle is equal to a distance of units. Therefore, a distance of units around the circle from the starting point would result in exactly full rotations, arriving back at the point . So, is equal to the y-coordinate of the point , which is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of , not .
Question 266 266 of 269 selected Right Triangles & Trigonometry
A triangle with angle measures 30°, 60°, and 90° has a perimeter of . What is the length of the longest side of the triangle?
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The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a special right triangle. The side measures of this type of special triangle are in the ratio . If x is the measure of the shortest leg, then the measure of the other leg is
and the measure of the hypotenuse is 2x. The perimeter of the triangle is given to be
, and so the equation for the perimeter can be written as
. Combining like terms and factoring out a common factor of x on the left-hand side of the equation gives
. Rewriting the right-hand side of the equation by factoring out 6 gives
. Dividing both sides of the equation by the common factor
gives x = 6. The longest side of the right triangle, the hypotenuse, has a length of 2x, or 2(6), which is 12.
Question 267 267 of 269 selected Area & Volume
A cube has a surface area of 54 square meters. What is the volume, in cubic meters, of the cube?
18
27
36
81
Show Answer
Correct Answer: BChoice B is correct. The surface area of a cube with side length s is equal to . Since the surface area is given as 54 square meters, the equation
can be used to solve for s. Dividing both sides of the equation by 6 yields
. Taking the square root of both sides of this equation yields
and
. Since the side length of a cube must be a positive value,
can be discarded as a possible solution, leaving
. The volume of a cube with side length s is equal to
. Therefore, the volume of this cube, in cubic meters, is
, or 27.
Choices A, C, and D are incorrect and may result from calculation errors.
Question 268 268 of 269 selected Area & Volume
In the xy-plane shown, square ABCD has its diagonals on the x- and y-axes. What is the area, in square units, of the square?
20
25
50
100
Show Answer
Correct Answer: CChoice C is correct. The two diagonals of square ABCD divide the square into 4 congruent right triangles, where each triangle has a vertex at the origin of the graph shown. The formula for the area of a triangle is , where b is the base length of the triangle and h is the height of the triangle. Each of the 4 congruent right triangles has a height of 5 units and a base length of 5 units. Therefore, the area of each triangle is
, or 12.5 square units. Since the 4 right triangles are congruent, the area of each is
of the area of square ABCD. It follows that the area of the square ABCD is equal to
, or 50 square units.
Choices A and D are incorrect and may result from using 5 or 25, respectively, as the area of one of the 4 congruent right triangles formed by diagonals of square ABCD. However, the area of these triangles is 12.5. Choice B is incorrect and may result from using 5 as the length of one side of square ABCD. However, the length of a side of square ABCD is .
Question 269 269 of 269 selected Circles
In the xy-plane, the graph of the equation is a circle. The point , where is a constant, lies on this circle. What is the value of ?
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Correct Answer: 5The correct answer is . It's given that in the xy-plane, the graph of the equation is a circle. It’s also given that the point , where is a constant, lies on this circle. It follows that the ordered pair makes the equation true. Substituting for and for in this equation yields , or . Subtracting from each side of this equation yields . It follows that the value of is .